cp's OEIS Frontend

A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

%I A384998 #22 Aug 12 2025 15:30:36
%S A384998 1,2,5,10,20,35,63,104,173,275,435,666,1018,1516,2248,3275,4745,6776,
%T A384998 9632,13528,18910,26182,36078,49311,67111,90690,122052,163271,217559,
%U A384998 288350,380806,500504,655601,855113,1111777,1439911,1859347,2392509,3069921,3926494
%N A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.
%H A384998 Alois P. Heinz, <a href="/A384998/b384998.txt">Table of n, a(n) for n = 0..10000</a>
%F A384998 From _Vaclav Kotesovec_, Aug 08 2025: (Start)
%F A384998 a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (2^(9/4) * sqrt(3) * Pi * n^(3/4)).
%F A384998 G.f.: 1/(1-x) * Product_{k>=1} (1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))). (End)
%p A384998 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A384998       b(n, i-1)+add(b(n-i*j, i-1)*j, j=1..n/i)))
%p A384998     end:
%p A384998 a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
%p A384998 seq(a(n), n=0..41);  # _Alois P. Heinz_, Aug 06 2025
%t A384998 nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 08 2025 *)
%Y A384998 Partial sums of A077285.
%Y A384998 Row sums of A384999.
%Y A384998 Cf. A385001.
%K A384998 nonn
%O A384998 0,2
%A A384998 _Omar E. Pol_, Aug 06 2025