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A320690 Number of partitions of n with up to three distinct kinds of 1. 2
1, 3, 4, 5, 8, 12, 17, 24, 33, 45, 61, 81, 107, 141, 183, 236, 304, 388, 492, 622, 782, 979, 1221, 1515, 1874, 2312, 2840, 3477, 4247, 5171, 6278, 7604, 9185, 11068, 13308, 15963, 19108, 22828, 27213, 32378, 38457, 45592, 53955, 63748, 75193, 88553, 104130 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

a(n) ~ Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - Vaclav Kotesovec, Oct 24 2018

G.f.: (1 + x)^3 * Product_{k>=2} 1 / (1 - x^k). - Ilya Gutkovskiy, Apr 24 2021

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1,

      binomial(3, n), `if`(i>n, 0, b(n-i, i))+b(n, i-1))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..60);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, Binomial[3, n], If[i > n, 0, b[n - i, i]] + b[n, i - 1]];

a[n_] := b[n, n];

a /@ Range[0, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

CROSSREFS

Column k=3 of A292622.

Sequence in context: A030403 A034403 A215082 * A179070 A039020 A055742

Adjacent sequences:  A320687 A320688 A320689 * A320691 A320692 A320693

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Oct 19 2018

STATUS

approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)