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A106337 Number of ways of writing n as the sum of n triangular numbers.

%I A106337 #26 Oct 07 2020 05:48:59
%S A106337 1,1,1,4,13,31,82,253,757,2173,6341,18888,56266,167324,499773,1499059,
%T A106337 4503557,13546893,40824379,123233868,372472353,1127080252,3414310032,
%U A106337 10353722919,31425764410,95463814056,290222666436,882954212908,2688037654049,8188468874808
%N A106337 Number of ways of writing n as the sum of n triangular numbers.
%C A106337 Number of compositions of n into n triangular numbers with 0's allowed.  a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - _Alois P. Heinz_, Jul 31 2017
%C A106337 The radius of convergence is equal to A106335. - _Vaclav Kotesovec_, Nov 15 2017
%H A106337 Alois P. Heinz, <a href="/A106337/b106337.txt">Table of n, a(n) for n = 0..2000</a>
%F A106337 Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).
%F A106337 a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - _Alois P. Heinz_, Aug 01 2017
%e A106337 G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
%e A106337 G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
%e A106337 G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...
%p A106337 b:= proc(n) option remember; expand(`if`(n=0, 1,
%p A106337       add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
%p A106337     end:
%p A106337 a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
%p A106337 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 31 2017
%t A106337 QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jun 04 2017 *)
%o A106337 (PARI) {a(n)=local(X); if(n<1,1,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n,n))}
%Y A106337 Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.
%Y A106337 Main diagonal of A286180.
%K A106337 nonn
%O A106337 0,4
%A A106337 _Paul D. Hanna_, Apr 29 2005
%E A106337 a(0) changed to 1 by _Alois P. Heinz_, Jul 31 2017