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 A005726 Quadrinomial coefficients. (Formerly M1643) 2
 1, 2, 6, 20, 65, 216, 728, 2472, 8451, 29050, 100298, 347568, 1208220, 4211312, 14712960, 51507280, 180642391, 634551606, 2232223626, 7862669700, 27727507521, 97884558992, 345891702456, 1223358393120, 4330360551700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 R. K. Guy, Letter to N. J. A. Sloane, 1987 FORMULA a(n) = Sum_{k=0..floor(n/2)}, C(n,k) C(n,2k+1). - Paul Barry, May 15 2003 a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-4k,n-1],{k,0,Floor[(n-1)/4]}]. - David Callan, Jul 03 2006 G.f.: F(G^(-1)(x)) where F(t) = (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1) and G(t) = t/((t-1)*(t^2+1)). - Mark van Hoeij, Oct 30 2011 Conjecture: 2*(n-1)*(2*n+1)*(13*n-14)*a(n) +(-143*n^3+297*n^2-148*n+12) *a(n-1) -4*(n-1)*(26*n^2-41*n+9)*a(n-2) -16*(n-1)*(n-2)*(13*n-1) *a(n-3)=0. - R. J. Mathar, Nov 13 2012 a(n) = A008287(n,n-1). - Sean A. Irvine, Aug 15 2016 MAPLE for n from 1 to 40 do printf(`%d, `, coeff(expand(sum(x^j, j=0..3)^n), x, n-1)) od: F := (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1);  G := t/((t-1)*(t^2+1)); Ginv := RootOf(numer(G-x), t);  ogf := series(eval(F, t=Ginv), x=0, 20); # Mark van Hoeij, Oct 30 2011 MATHEMATICA Table[Sum[Binomial[n, k]Binomial[n, 2k+1], {k, 0, Floor[n/2]}], {n, 30}] (* Harvey P. Dale, Oct 19 2013 *) CROSSREFS Sequence in context: A273902 A181301 A302612 * A148473 A000718 A148474 Adjacent sequences:  A005723 A005724 A005725 * A005727 A005728 A005729 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from James A. Sellers, Aug 21 2000 STATUS approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)