

A334461


a(n) is the number of partitions of n into consecutive parts that differ by 4.


10



1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 5, 1, 2, 2, 3, 2, 4, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 1, 4, 1, 4, 2, 2, 1, 5, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 5, 1, 3, 2, 4, 1, 4, 1, 3, 4
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OFFSET

1,6


LINKS



FORMULA

The g.f. for "consecutive parts that differ by d" is Sum_{k>=1} x^(k*(d*kd+2)/2) / (1x^k); cf. A117277.  Joerg Arndt, Nov 30 2020


EXAMPLE

For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a valid partition. They are [28], [16, 12] and [13, 9, 5, 1]. So a(28) = 3.


MATHEMATICA

nmax = 105;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n  k + 2)/2  1)/(1  x^n), {n, 1, nmax}] + O[x]^nmax, x];
a[n_] := col[4][[n]];


PROG

(PARI) seq(N, d)=my(x='x+O('x^N)); Vec(sum(k=1, N, x^(k*(d*kd+2)/2)/(1x^k)));


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



