A325434 - OEIS

%I #14 Feb 22 2021 03:36:47

%S 0,1,1,2,2,4,5,8,10,15,19,27,34,47,60,80,101,133,167,216,270,344,428,

%T 540,667,834,1026,1271,1555,1914,2330,2849,3453,4197,5065,6125,7360,

%U 8858,10605,12706,15155,18086,21497,25557,30279,35870,42366,50026,58909,69346

%N Row sums of A325433.

%H Vaclav Kotesovec, <a href="/A325434/b325434.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

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

%t 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 *)

%t Table[Sum[T[n,k],{k,1,n}],{n,1,50}]

%o (PARI)

%o 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

%o a(n) = sum(k=1, n, T(n,k));

%Y Cf. A000041, A002865, A325433.

%K nonn

%O 1,4

%A _Stefano Spezia_, Apr 27 2019