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
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
Programs
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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
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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
Comments