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

0,2

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A037028 A052919 A005807 * A060276 A337187 A358049

Adjacent sequences: A167419 A167420 A167421 * A167423 A167424 A167425

Keyword

easy,nonn

Author

Paul Barry, Nov 03 2009

Status

approved