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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

GAP

List([0..30], 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

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 *)

PARI

vector(30, n, n--; binomial(2*n+6, n+3)*binomial(n+3, 3)/20 ) \\ 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

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