A167422 Expansion of (1+x)*c(x), c(x) the g.f. of A000108.

1, 2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798, 950912, 3417340, 12369285, 45052515, 165002460, 607283490, 2244901890, 8331383610, 31030387440, 115948830660, 434542177290, 1632963760974, 6151850548776

Offset: 0

Author

Paul Barry, Nov 03 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A167423.

Apparently a(n) = A071716(n) if n>1. - R. J. Mathar, Nov 12 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2

Crossrefs

Cf. A000108, A001450, A005807, A071716, A167423, A182959.

Programs

Maple

A167422List := proc(m) local A, P, n; A := [1, 2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), A[-1]]);
A := [op(A), P[-1]] od; A end: A167422List(26); # Peter Luschny, Mar 24 2022

Mathematica

Table[If[n < 2, n + 1, Binomial[2 n, n]/(n + 1) + Binomial[2 (n - 1), n - 1]/n], {n, 0, 25}] (* Michael De Vlieger, Oct 05 2015 *)
CoefficientList[Series[(1 + t)*(1 - Sqrt[1 - 4*t])/(2*t), {t, 0, 50}], t] (* G. C. Greubel, Jun 12 2016 *)

PARI

a(n) = if (n<2, n+1, binomial(2*n, n)/(n+1) + binomial(2*(n-1), n-1)/n);
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Formula

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

a(0)= 1, a(n) = A005807(n-1) for n>0. - Philippe Deléham, Nov 25 2009

(n+1)*a(n) +(-3*n+1)*a(n-1) +2*(-2*n+5)*a(n-2)=0, for n>2. - R. J. Mathar, Feb 10 2015

-(n+1)*(5*n-6)*a(n) +2*(5*n-1)*(2*n-3)*a(n-1)=0. - R. J. Mathar, Feb 10 2015

The o.g.f. A(x) satisfies [x^n] A(x)^(5*n) = binomial(5*n,2*n) = A001450(n). Cf. A182959. - Peter Bala, Oct 04 2015