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

%I A167422 #33 Mar 24 2022 18:52:14
%S A167422 1,2,3,7,19,56,174,561,1859,6292,21658,75582,266798,950912,3417340,
%T A167422 12369285,45052515,165002460,607283490,2244901890,8331383610,
%U A167422 31030387440,115948830660,434542177290,1632963760974,6151850548776
%N A167422 Expansion of (1+x)*c(x), c(x) the g.f. of A000108.
%C A167422 Hankel transform is A167423.
%C A167422 Apparently a(n) = A071716(n) if n>1. - _R. J. Mathar_, Nov 12 2009
%H A167422 G. C. Greubel, <a href="/A167422/b167422.txt">Table of n, a(n) for n = 0..500</a>
%H A167422 Murray Tannock, <a href="https://skemman.is/bitstream/1946/25589/1/msc-tannock-2016.pdf">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2
%F A167422 a(n) = Sum_{k=0..n} A000108(k)*C(1,n-k).
%F A167422 a(0)= 1, a(n) = A005807(n-1) for n>0. - _Philippe Deléham_, Nov 25 2009
%F A167422 (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
%F A167422 -(n+1)*(5*n-6)*a(n) +2*(5*n-1)*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Feb 10 2015
%F A167422 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
%p A167422 A167422List := proc(m) local A, P, n; A := [1, 2]; P := [1];
%p A167422 for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), A[-1]]);
%p A167422 A := [op(A), P[-1]] od; A end: A167422List(26); # _Peter Luschny_, Mar 24 2022
%t A167422 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 *)
%t A167422 CoefficientList[Series[(1 + t)*(1 - Sqrt[1 - 4*t])/(2*t), {t, 0, 50}], t] (* _G. C. Greubel_, Jun 12 2016 *)
%o A167422 (PARI) a(n) = if (n<2, n+1, binomial(2*n, n)/(n+1) + binomial(2*(n-1), n-1)/n);
%o A167422 vector(50, n, a(n-1)) \\ _Altug Alkan_, Oct 04 2015
%Y A167422 Cf. A000108, A001450, A005807, A071716, A167423, A182959.
%K A167422 easy,nonn
%O A167422 0,2
%A A167422 _Paul Barry_, Nov 03 2009