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

1, 2, 5, 10, 20, 35, 63, 104, 173, 275, 435, 666, 1018, 1516, 2248, 3275, 4745, 6776, 9632, 13528, 18910, 26182, 36078, 49311, 67111, 90690, 122052, 163271, 217559, 288350, 380806, 500504, 655601, 855113, 1111777, 1439911, 1859347, 2392509, 3069921, 3926494

Offset: 0

Author

Omar E. Pol, Aug 06 2025

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Partial sums of A077285.

Row sums of A384999.

Cf. A385001.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..41);  # Alois P. Heinz, Aug 06 2025

Mathematica

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 *)

Formula

From Vaclav Kotesovec, Aug 08 2025: (Start)

a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (2^(9/4) * sqrt(3) * Pi * n^(3/4)).

G.f.: 1/(1-x) * Product_{k>=1} (1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))). (End)