A020918 Expansion of 1/(1-4*x)^(7/2).

1, 14, 126, 924, 6006, 36036, 204204, 1108536, 5819814, 29745716, 148728580, 730122120, 3528923580, 16830250920, 79342611480, 370265520240, 1712478031110, 7857252142740, 35794148650260

Offset

0,2

Comments

Also convolution of A000984 with A038845, also convolution of A000302 with A002802, also convolution of A002457 with A002697. - Rui Duarte, Oct 08 2011

5*a(n) is the number of (n+3) X 2 Young tableaux with a four horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021], A000984 for one horizontal wall, A002457 for two, and A002802 for three. - Michael Wallner, Mar 09 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.

Formula

a(n) = binomial(n+3, 3)*A000984(n+3)/A000984(3), where A000984 are the central binomial coefficients. - Wolfdieter Lang

a(n) ~ 8/15*Pi^(-1/2)*n^(5/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001

a(n) = Sum_{a+b+c+d+e+f+g=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004

a(n) = A000292(n)*A000984(n+2)/20. - Zerinvary Lajos, May 05 2007

From Rui Duarte, Oct 08 2011: (Start)

a(n) = ((2n+5)(2n+3)(2n+1)/(5*3*1)) * binomial(2n, n).

a(n) = binomial(2n+6, 6) * binomial(2n, n) / binomial(n+3, 3).

a(n) = binomial(n+3, 3) * binomial(2n+6, n+3) / binomial(6, 3). (End)

a(n) = 4^n*hypergeom([-n,-5/2], [1], 1). - Peter Luschny, Apr 26 2016

Boas-Buck recurrence: a(n) = (14/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+3, 3). See a comment there. - Wolfdieter Lang, Aug 10 2017

From Amiram Eldar, Apr 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 160/3.

Sum_{n>=0} (-1)^n/a(n) = 10*sqrt(5)*log(phi) - 320/3, where phi is the golden ratio (A001622). (End)

D-finite with recurrence n*a(n) +2*(-2*n-5)*a(n-1)=0. - R. J. Mathar, Aug 01 2022

Maple

seq(binomial(2*n, n)*binomial(n, (n-3))/20, n=2..21); # Zerinvary Lajos, May 05 2007
seq(simplify(4^n*hypergeom([-n, -5/2], [1], 1)), n=0..18); # Peter Luschny, Apr 26 2016

Mathematica

CoefficientList[Series[1/(1-4x)^(7/2), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 04 2013 *)

Prog

(PARI) vector(30, n, n--; binomial(2*n+6, n+3)*binomial(n+3, 3)/20 ) \\ G. C. Greubel, Jul 20 2019
(Magma) [Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20: n in [0..30]]; // G. C. Greubel, Jul 20 2019
(Sage) [binomial(2*n+6, n+3)*binomial(n+3, 3)/20 for n in (0..30)] # G. C. Greubel, Jul 20 2019
(GAP) List([0..30], n-> Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20); # G. C. Greubel, Jul 20 2019

Crossrefs

Cf. A000302, A000984, A001622, A002457, A002697, A002802, A000292, A038845, A046521 (fourth column).

Sequence in context: A090296 A088625 A073393 * A275559 A222477 A041368

Adjacent sequences: A020915 A020916 A020917 * A020919 A020920 A020921

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved