A155051 Expansion of c(x^2)*(1+x)/(1-x), c(x) the g.f. of A000108.

1, 2, 3, 4, 6, 8, 13, 18, 32, 46, 88, 130, 262, 394, 823, 1252, 2682, 4112, 8974, 13836, 30632, 47428, 106214, 165000, 373012, 581024, 1323924, 2066824, 4741264, 7415704, 17110549, 26805394, 62163064, 97520734, 227165524

Offset: 0

Author

Paul Barry, Jan 19 2009

Keywords

• easy
• nonn

Comments

Row sums of A155050.

Conjecture: A000975(n) = A264784(a(n-1)) for n > 0. - Reinhard Zumkeller, Dec 04 2015

Links

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

Crossrefs

Cf. A000108, A155050, A000975, A264784.

Programs

Mathematica

A155051[n_] := 2*Sum[CatalanNumber[k/2]*(1 + (-1)^k)/2, {k, 0, n}] -
CatalanNumber[n/2]*(1 + (-1)^n)/2; Table[A155051[n], {n, 0, 50}] (* G. C. Greubel, Sep 30 2017 *)

Formula

a(n) = 2*Sum_{k=0..n,} ( C(k/2)*(1+(-1)^k)/2 ) - C(n/2)*(1+(-1)^n)/2, C(n) = A000108;

a(n) = (C(n/2) + 2*Sum_{k=0..(n/2-1), C(k)})*(1+(-1)^n)/2 + Sum_{k=0..n/2, C(k)}*(1-(-1)^n), C(n) = A000108.

Conjecture: (n+2)*a(n) -2*a(n-1) +(-5*n+4)*a(n-2) +8*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 05 2015

Conjecture: -(n+2)*(n-3)*a(n) +(n^2-n-10)*a(n-1) +4*(n^2-4*n+5)*a(n-2) -4*(n-2)^2*a(n-3)=0. - R. J. Mathar, Feb 05 2015