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A020918 Expansion of 1/(1-4*x)^(7/2). 13
1, 14, 126, 924, 6006, 36036, 204204, 1108536, 5819814, 29745716, 148728580, 730122120, 3528923580, 16830250920, 79342611480, 370265520240, 1712478031110, 7857252142740, 35794148650260 (list; graph; refs; listen; history; text; internal format)
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

LINKS

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

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

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, 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

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Last modified May 18 22:15 EDT 2021. Contains 344003 sequences. (Running on oeis4.)