cp's OEIS Frontend

A106336 Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.

%I A106336 #39 Jun 12 2025 18:59:11
%S A106336 1,1,1,2,5,11,25,64,169,442,1172,3180,8730,24116,67159,188568,532741,
%T A106336 1512695,4315996,12369324,35587923,102747636,297601382,864525312,
%U A106336 2518185362,7353088206,21520084301,63115752910,185474840912,546042990300,1610314638958
%N A106336 Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.
%C A106336 Apparently: Number of Dyck n-paths with each ascent length being a triangular number. - _David Scambler_, May 09 2012
%H A106336 Alois P. Heinz, <a href="/A106336/b106336.txt">Table of n, a(n) for n = 0..2000</a>
%F A106336 G.f.: A(x) = (1/x) * Series_Reversion( x*eta(x)/eta(x^2)^2 ).
%F A106336 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A106336 (1) A(x) = F(x*A(x)) where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
%F A106336 (2) log(A(x)) = Sum_{n>=1} A106337(n)/n*x^n.
%F A106336 (3) A(x) = Product_{n>=1} (1 + (x*A(x))^n)*(1 - (x*A(x))^(2*n)). - _Paul D. Hanna_, Oct 23 2010
%F A106336 (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 A106336 From _Paul D. Hanna_, Jun 11 2025: (Start)
%F A106336 (5) A(x)^4 = Sum_{n>=0} (2*n+1) * (x*A(x))^n / (1 - (x*A(x))^(2*n+1)).
%F A106336 (6) A(x^2)^2 = Sum_{n>=0} (x*A(x^2)^(1/2))^n / (1 + (x*A(x^2)^(1/2))^(2*n+1)).
%F A106336 (End)
%F A106336 a(n) ~ c / (n^(3/2) * A106335^n), where c = A366174 = 0.49833479793360342260635926402850016443069428233051290201996853498... - _Vaclav Kotesovec_, Oct 07 2020
%e A106336 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 64*x^7 +...
%e A106336 A(x) = F(x*A(x)) where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + ...
%e A106336 The radius of convergence equals r = 0.322627632692191133... (A106335)
%e A106336 at which the g.f. converges to A(r) = 1.987369721184684145... (A106334).
%p A106336 b:= proc(n) option remember; expand(`if`(n=0, 1,
%p A106336       add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
%p A106336     end:
%p A106336 a:= n-> (p-> add(coeff(p, x, i)*binomial(1+n, i),
%p A106336              i=0..n)/(n+1))(b(n)):
%p A106336 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jul 31 2017
%t A106336 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 A106336 (PARI) {a(n) = my(X); if(n<0,0,X=x+x*O(x^n); polcoef(eta(X^2)^(2*n+2)/eta(X)^(n+1)/(n+1),n))}
%o A106336 (PARI) {a(n) = if(n<0,0,polcoef( sum(k=1,(sqrtint(8*n+1)+1)\2,x^((k^2-k)/2),x*O(x^n))^(n+1)/(n+1),n))}
%o A106336 (PARI) {a(n) = my(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))));polcoef(A,n)} \\ _Paul D. Hanna_, Oct 23 2010
%o A106336 (PARI) {a(n) = my(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)));polcoef(A,n)} \\ _Paul D. Hanna_, Jun 01 2011
%Y A106336 Cf. A106333, A106334, A106335, A106337; related: A109085.
%K A106336 nonn
%O A106336 0,4
%A A106336 _Paul D. Hanna_, Apr 29 2005
%E A106336 Edited by _Paul D. Hanna_, Jun 01 2011