A308264 Number of ordered factorizations of n into triangular numbers > 1.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 1

Author

Ilya Gutkovskiy, May 17 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..101025

Crossrefs

Cf. A000217, A010054, A085782 (for n > 1, positions of nonzero terms).

Not the absolute values of A365800.

Programs

Mathematica

terms = 100; A[] = 0; Do[A[x] = x + Sum[A[x^(k (k + 1)/2)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[(d < n && IntegerQ[Sqrt[8 n/d + 1]]), a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 100}]

PARI

A010054(n) = issquare(1 + 8*n);
A308264(n) = if(1==n,n,sumdiv(n,d,if(dA010054(n/d)*A308264(d),0))); \\ Antti Karttunen, Oct 05 2023

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} A(x^(k*(k+1)/2)).

a(1) = 1; a(n) = Sum_{d|n, dA010054(n/d)*a(d).

Extensions

Data section extended to 105 terms by Antti Karttunen, Oct 05 2023