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%I #33 Oct 15 2023 19:06:00
%S 1,1,1,2,5,11,25,64,169,442,1172,3180,8730,24116,67159,188568,532741,
%T 1512695,4315996,12369324,35587923,102747636,297601382,864525312,
%U 2518185362,7353088206,21520084301,63115752910,185474840912,546042990300,1610314638958
%N Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.
%C Apparently: Number of Dyck n-paths with each ascent length being a triangular number. - _David Scambler_, May 09 2012
%H Alois P. Heinz, <a href="/A106336/b106336.txt">Table of n, a(n) for n = 0..2000</a>
%F G.f.: A(x) = (1/x) * Series_Reversion( x*eta(x)/eta(x^2)^2 ).
%F G.f. satisfies:
%F (1) A(x) = F(x*A(x)) where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
%F (2) log(A(x)) = Sum_{n>=1} A106337(n)/n*x^n.
%F (3) A(x) = Product_{n>=1} (1 + (x*A(x))^n)*(1 - (x*A(x))^(2n)). - _Paul D. Hanna_, Oct 23 2010
%F (4) A(x) = exp( Sum_{n>=1} (x^n*A(x)^n/(1 + x^n*A(x)^n))/n ). - _Paul D. Hanna_, Jun 01 2011
%F a(n) ~ c / (n^(3/2) * A106335^n), where c = A366174 = 0.49833479793360342260635926402850016443069428233051290201996853498... - _Vaclav Kotesovec_, Oct 07 2020
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 64*x^7 +...
%e A(x) = F(x*A(x)) where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + ...
%e The radius of convergence equals r = 0.322627632692191133... (A106335)
%e at which the g.f. converges to A(r) = 1.987369721184684145... (A106334).
%p b:= proc(n) option remember; expand(`if`(n=0, 1,
%p add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
%p end:
%p a:= n-> (p-> add(coeff(p, x, i)*binomial(1+n, i),
%p i=0..n)/(n+1))(b(n)):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jul 31 2017
%t f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x-1, y, 0] + f[x, y - If[d == 0, 1, Ceiling[Sqrt[2*d]]],If[d == 0, 1, Ceiling[Sqrt[2*d]] + d]]]]; Table[f[n, n, 0], {n, 0, 30}] (* _David Scambler_, May 09 2012 *)
%o (PARI) {a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n+2)/eta(X)^(n+1)/(n+1),n))}
%o (PARI) {a(n)=if(n<0,0,polcoeff( sum(k=1,(sqrtint(8*n+1)+1)\2,x^((k^2-k)/2),x*O(x^n))^(n+1)/(n+1),n))}
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=prod(m=1,n,(1+(x*A)^m)*(1-(x*A)^(2*m))));polcoeff(A,n)} \\ _Paul D. Hanna_, Oct 23 2010
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(x*A)^m/(1+(x*A)^m+x*O(x^n))/m)));polcoeff(A,n)} \\ _Paul D. Hanna_, Jun 01 2011
%Y Cf. A106333, A106334, A106335, A106337; related: A109085.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Apr 29 2005
%E Edited by _Paul D. Hanna_, Jun 01 2011
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