A136520 a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

1, 1, 3, 13, 44, 146, 530, 1975, 7314, 27262, 102802, 390138, 1486064, 5682756, 21812436, 83976075, 324115550, 1253795510, 4859960402, 18871869302, 73398851448, 285882923196, 1114943553308, 4353426835238, 17016813133124, 66581653586476, 260750813149140, 1022023318047220

Offset: 1

Author

Gary W. Adamson, Jan 02 2008

Keywords

• nonn
• easy

Comments

Narayana transform of A027656.

Examples

			a(4) = 13 = (1, 6, 6, 1) dot (1, 0, 2, 0) = (1 + 0 + 12 + 0).
Triangle A001263(n,k) * A027656(k+1) and the rows sums:
  1;                                              :     1;
  1,  0;                                          :     1;
  1,  0,    2;                                    :     3;
  1,  0,   12,  0;                                :    13;
  1,  0,   40,  0,     3;                         :    44;
  1,  0,  100,  0,    45,  0;                     :   146;
  1,  0,  210,  0,   315,  0,     4;              :   530;
  1,  0,  392,  0,  1470,  0,   112,  0;          :  1975;
  1,  0,  672,  0,  5292,  0,  1344,  0,    5;    :  7314;
  1,  0, 1080,  0, 15876,  0, 10080,  0,  225,  0 : 27262;
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001263, A027656.

Programs

Magma

A136520:= func< n | (&+[((j+1)/(2*j+1))*Binomial(n,2*j)*Binomial(n-1,2*j): j in [0..Floor((n-1)/2)]]) >;
[A136520(n): n in [1..40]]; // G. C. Greubel, Jul 27 2023

Mathematica

A136520[n_]:= Sum[Binomial[n-1, 2*k]*Binomial[n, 2*k]*((k+1)/(2*k+1)), {k,0,Floor[(n-1)/2]}];
Table[A136520[n], {n, 40}] (* G. C. Greubel, Jul 27 2023 *)

SageMath

def A136520(n): return sum(((j+1)/(2*j+1))*binomial(n,2*j)*binomial(n-1, 2*j) for j in range((n+1)//2))
[A136520(n) for n in range(1,41)] # G. C. Greubel, Jul 27 2023

Formula

a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

a(n) = Sum_{j=0..floor((n-1)/2)} ((j+1)/(2*j+1))*binomial(n, 2*j) * binomial(n-1, 2*j). - G. C. Greubel, Jul 27 2023

Extensions

Terms a(11) onward added by G. C. Greubel, Jul 27 2023