|
| |
|
|
A005726
|
|
Quadrinomial coefficients.
(Formerly M1643)
|
|
1
|
|
|
|
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
|
Table of n, a(n) for n=1..25.
|
|
|
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
|
|
|
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
|
|
|
CROSSREFS
|
Sequence in context: A053730 A220874 A181301 * A148473 A000718 A148474
Adjacent sequences: A005723 A005724 A005725 * A005727 A005728 A005729
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
EXTENSIONS
|
More terms from James A. Sellers, Aug 21 2000
|
|
|
STATUS
|
approved
|
| |
|
|
