A098132 -id:A098132 - cp's OEIS frontend

A321481 Expansion of Sum_{n>=1} q^(n*(n-1)) / (1-q)^n.

1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 25, 32, 41, 53, 69, 90, 117, 151, 193, 244, 306, 382, 476, 593, 739, 921, 1147, 1426, 1768, 2184, 2687, 3293, 4022, 4899, 5955, 7228, 8764, 10618, 12855, 15551, 18794, 22685, 27340, 32893, 39500, 47344, 56641, 67647, 80666, 96059, 114254, 135757, 161164, 191174, 226603, 268399

Offset: 0

Joerg Arndt, Nov 11 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A098132 (expansion of Sum_{n>=0} q^(n*(n+1)) / (1-q)^n ).

Cf. A063978.

nmax = 60; CoefficientList[Series[Sum[x^(k*(k-1))/(1-x)^k, {k, 1, Sqrt[nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 11 2018 *)

N=66; q='q+O('q^N); Vec( sum(n=1,N,q^(n*(n-1))/(1-q)^n) )

G.f.: Sum_{n>=1} q^(n*(n-1)) / (1-q)^n.

A348124 Number of compositions of n where the smallest part is smaller than the number of parts.

Original entry on oeis.org

0, 1, 3, 6, 13, 28, 59, 122, 248, 501, 1009, 2028, 4070, 8159, 16343, 32717, 65472, 130991, 262041, 524157, 1048410, 2096943, 4194043, 8388285, 16776819, 33553946, 67108270, 134217002, 268434568, 536869825, 1073740493, 2147482019, 4294965305, 8589932164

Offset: 1

R. J. Mathar, Oct 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A011782, A098132, A098133.

b:= proc(n, s, c) option remember; `if`(s b(n$2, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 01 2021

b[n_, s_, c_] := b[n, s, c] = If[s < c, Ceiling[2^(n - 1)],
     If[n == 0, 0, Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

a(n) + A098132(n) + A098133(n) = 2^(n-1).

a(23)-a(34) from Alois P. Heinz, Oct 01 2021

cp's OEIS Frontend

A321481 Expansion of Sum_{n>=1} q^(n*(n-1)) / (1-q)^n.

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