A325434 Row sums of A325433.

0, 1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 60, 80, 101, 133, 167, 216, 270, 344, 428, 540, 667, 834, 1026, 1271, 1555, 1914, 2330, 2849, 3453, 4197, 5065, 6125, 7360, 8858, 10605, 12706, 15155, 18086, 21497, 25557, 30279, 35870, 42366, 50026, 58909, 69346

Offset: 1

Author

Stefano Spezia, Apr 27 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000041, A002865, A325433.

Programs

Mathematica

T[n_,k_]:=(-1)^(k-1)*Sum[(-1)^j*(PartitionsP[n-j*(3*j+1)/2]-PartitionsP[n-j*(3*j+5)/2-1]),{j,0,k-1}]; (* A325433 *)
Table[Sum[T[n,k],{k,1,n}],{n,1,50}]

PARI

T(n,k) = (-1)^(k-1)*sum(j=0, k-1, (-1)^j*(numbpart(n-j*(3*j+1)/2)-numbpart(n-j*(3*j+5)/2-1))); \\ A325433
a(n) = sum(k=1, n, T(n,k));

Formula

a(n) = Sum_{k=1..n} ((-1)^(k-1)*Sum_{j=0..k-1} (-1)^j*(p(n - j*(3*j + 1)/2) - p(n - j*(3*j + 5)/2 - 1))), where p(n) = A000041(n) is the number of partitions of n.

Conjecture: Lim_{n->infinity} a(n)/A000041(n) = 1/3.