A027277 a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).

1, 3, 13, 67, 375, 2189, 13089, 79479, 487833, 3018355, 18792303, 117589689, 738844719, 4658460165, 29458662005, 186761788579, 1186655988771, 7554520173441, 48176764031385, 307706150625855, 1968040844127793, 12602972755261195, 80798365998084795, 518536437750443773

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Previous name was: a(n) = self-convolution of row n of array T given by A026568.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A026568.

Programs

GAP

B:=Binomial;; List([0..30], n-> Sum([0..n], k-> B(2*k,k)*B(2*n-k,k) )); # G. C. Greubel, Aug 03 2019

Magma

B:=Binomial; [(&+[B(2*k,k)*B(2*n-k,k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 03 2019

Maple

a := n -> add(binomial(2*k,k)*binomial(2*n-k,k), k=0..n):
seq(a(n),n=0..23); # Peter Luschny, May 14 2016

Mathematica

Table[Sum[Binomial[2k, k] Binomial[2n-k, k], {k,0,n}], {n,0,30}] (* Michael De Vlieger, May 14 2016 *)

PARI

vector(30, n, n--; b=binomial; sum(k=0,n, b(2*k,k)*b(2*n-k,k)) ) \\ G. C. Greubel, May 23 2017, modified Aug 03 2019

Sage

b=binomial; [sum(b(2*k,k)*b(2*n-k,k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Aug 03 2019

Formula

From Peter Luschny, May 14 2016: (Start)

a(n) = hypergeom([1/2, -n, 1/2-n], [1, -2*n], -16) for n>=1.

a(n) = (2*n*(4*n-5)*(-9+4*n)*(-7+4*n)*a(n-3) - (4*n-5)*(50*n^3-175*n^2+152*n-9)* a(n-2) + (80*n^3-260*n^2+198*n-27)*(n-1)*a(n-1)) / (n*(n-1)*(-9+4*n)*(-1+2*n)) for n>=3. (End)

a(n) ~ sqrt(5 + 13/sqrt(17)) * ((9 + sqrt(17))/2)^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 14 2016

Extensions

New name from Peter Luschny, May 14 2016