How to prove the following combinatorial identity-2?
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From: leox on 21 Dec 2009 03:52 On 21 çÒÄ, 05:29, achille <achille_...(a)yahoo.com.hk> wrote: > On Dec 21, 5:21am, leox <leonid...(a)gmail.com> wrote: > > > 1/((1-x)(1-y) (1-xy))=\sum_{i,j=0} min(i+1,j+1) x^i y^j > > Same trick, split the sum into two cases > i <= j (introduce index l = 0..\infty : j = i+l ) > and j < i (introduce index m = 1..\infty : i = j+m ) to get > > \sum_{i=0} (i+1) (xy)^i \sum{l=0} y^l > + \sum_{j=0} (j+1) (xy)^j \sum{m=1} x^m > > and simplify.... Thank you.
From: M. M i c h a e l M u s a t o v on 21 Dec 2009 08:30
http://www.meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A9%3B+NB%3A1&ie=UTF-16&q=%3E+%9A+%5Csum_%7Bi%3D0%7D+(i%2B1)+(xy)%5Ei+%5Csum%7Bl%3D0%7D+y%5El+#1232 a(n) = (-1/2) Sum (-3)^i C(1/2, i) C(1/2, j); i+j=n+2, i >= 0, j >= 0. a(n) = (3/2)^(n+2) * Sum_{k >= 1} 3^(-k) * Catalan(k-1) * binomial(k, n +2-k) [Doslic et al.] ... G.f.: A(x)=(1-y+y^2)/(1-y)^2 where (1+x)(y^2- y)+x=0; ... www.research.att.com/~njas/sequences/A001006 1 2 3 4 5 6 7 8 9 10 Next On Dec 21, 12:52 am, leox <leonid...(a)gmail.com> wrote: > On 21 çÃÃ, 05:29, achille <achille_...(a)yahoo.com.hk> wrote: > > > On Dec 21, 5:21Å¡am, leox <leonid...(a)gmail.com> wrote: > > > > 1/((1-x)(1-y) (1-xy))=\sum_{i,j=0} min(i+1,j+1) x^i y^j > > > Same trick, split the sum into two cases > > Å¡ Å¡ i <= j (introduce index l = 0..\infty : j = i+l ) > > and j Å¡< i (introduce index m = 1..\infty : i = j+m ) to get > > > Å¡ \sum_{i=0} (i+1) (xy)^i \sum{l=0} y^l > > + \sum_{j=0} (j+1) (xy)^j \sum{m=1} x^m > > > and simplify.... > > Thank you. Results 1 - 10 for > Å¡ \sum_{i=0} (i+1) (xy)^i \sum{l=0} y^l. (0. 20 seconds) 1. Result for query "keyword(s)=frac author= title=" \sum_ k=0^ m_e-1(\ \frac k+\gamma_e m_e\,0)-(\ \frac k+\gamma_e m_e\+1, . .. .. $$(x,y)_z=\frac 1 2(d(z,x)+d(z,y)-d(x,y)).$$ The quantity $(x,y)_z$ . .. .. Characterizations of Embeddable 3 x 3 Stochastic Matrices with a Negative Eigenvalue . .. .. \sum^\infty_ n=1 T^n_0 (I - T_0 T^*_0 )^ \frac 1 2 L; )^ \frac 1 2 L W_n f \ . .. .. nyjm. albany. edu:8000/cgi-bin/aglimpse/19/nyjm/Http/. .. .. /j?. .. .. 2. id:A000225 - OEIS Search Results 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, . .. .. .. a(n) = sum of previous terms + n = (Sum_(i=0. .. .. n-1) a(i)) + n for n >= 1. . .. .. .. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs. , Vol. . .. .. a(n) = n + sum(i=0, n-1, a(i)); a(0) = 0. - Rick L. Shepherd . .. .. www. research. att. com/~njas/sequences/A000225 3. Result for query "keyword(s)=frac author= title=" $$\lim_ n \to \infty \frac 1 n\sum_ i=0^ n-1[h_1(f^i(x))] . .. .. .. $$(x,y)_z=\frac 1 2(d(z,x)+d(z,y)-d(x,y)).$$ The quantity $(x,y)_z$; $d_ \Cal M$, . .. .. Characterizations of Embeddable 3 x 3 Stochastic Matrices with a Negative Eigenvalue . .. .. \sum^\infty_ n=1 T^n_0 (I - T_0 T^*_0 )^ \frac 1 2 L; )^ \frac 1 2 L W_n f \ . .. .. nyjm. albany. edu:8000/cgi-bin/aglimpse/19/nyjm/Http/. .. .. /j?. .. .. 4. id:A007318 - OEIS Search Results C(n-3,k-1) counts the permutations in S_n which have zero occurrences of the pattern 231 and one . .. .. .. L. Euler, On the expansion of the power of any polynomial (1+x +x^2+x^3+x^4+etc)^n . .. .. G. f. : 1/(1-y-xy)=Sum(C(n, k)x^k*y^n, n, k>=0); . .. .. of (n+1), sum_[p(i)=k]_{i=1}^{P(n+1)} = sum running from i=1 to i=P(n +1) but . .. .. www. research. att. com/~njas/sequences/A007318 5. Result for query "keyword(s)=substack author= title=" . .. .. \bl \qquad n_1 + \cdots +n_ k = m; \sum \limits_ \substack \bb \bl \qquad n_1 + \cdots +n_ k = m . .. .. E=\xi B\oplus\bigoplus_ \substack n\ge1\\; L^2(A,\Phi_ \iota_0)=A_ . .. .. \deg(X)= \displaystyle \sum_ \substack i=1,\dots,n \\ j=0,1,2,3 . .. .. h & \leq \frac \Sum_ \substack x \sim y \\x \in X, y \in S \setminus X . .. .. nyjm. albany. edu:8000/cgi-bin/aglimpse/19/nyjm/Http/. .. .. /j?. .. .. 2001. .. .. 6. id:A008277 - OEIS Search Results [From Roger L. Bagula (rlbagulatftn(AT)yahoo. com), Jan 11 2009] . .. .. .. Sum_{i=0. .. k} (-1)^(k-i)*C(k, i)*i^n. Bell number A000110(n) = sum(S(n, k)) k=1. .. n, n>0. . .. .. (1/3!) = 15. The sum of the complexions is 15+60+15=90=S2(6, 3). . .. .. Lag(n,x,m), the associated Laguerre polynomials of order m; and C (x,y) = x!/[ y! . .. .. www. research. att. com/~njas/sequences/A008277 7. Result for query "keyword(s)=substack author= title=" . .. .. \bar \delta_X(t)=\inf_ \|x\|=1\sup_ \dim X/Y<\infty\inf_ \substack y \in Y\\ . .. .. .. |\cl C^0|\mu(b_0) & = \sum_ b\in \cl C^0\sum_ \substack t\in \cl C^1 . .. .. |L|=q \sum _ \substack 1\leq j\leq p; \cal C_h(F^ \lambda _ _ J)= \sum_ . .. .. + \sum_ \substack h \in F' \setminus F \\ y \in \nSeq[n+1] k_n s_n^2 h^ -1 x_h . .. .. nyjm. albany. edu:8000/cgi-bin/aglimpse/19/nyjm/Http/. .. .. /j?. .. .. 8. id:A008292 - OEIS Search Results L. Carlitz et al. , Permutations and sequences with repetions by number of . .. .. .. T(i, n) = sum_{j=0}^{i} (-1)^j (n+1 combin j) (i-j+1)^n for n>=1, i>=0. . .. .. Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, ... y!*(x-y)! ]. For x = 0, the equation gives sum (j=0,...,n) E(n,j) * C(j,n) ... www.research.att.com/~njas/sequences/A008292 9. Result for query "keyword(s)=dfrac author= title=" ... \p x_3\in C_b(R^3)$,; u_ tt+\Delta^2u_t+\sum\limits_ i=1^ 3a_i(x)\dfrac \p \p x_i .... U(x,y)=\dfrac \partial U \partial n_x (x,y)=0; \dfrac \partial \partial n_y D(\xi ... $(t\in\C-0).$ It is the straight line $h=\dfrac c-1 24(1-k^2)$ if $k=l$ ... E_ 2n(q) =1- \dfrac 4n B_ 2n\sum_ k\geq 1\sigma_ 2n-1(k)q^k ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/.../j?... 10. id:A001006 - OEIS Search Results Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, ... |