A246937 Number of partitions of n into 5 sorts of parts.

1, 5, 30, 155, 805, 4055, 20455, 102455, 513230, 2567230, 12841130, 64211380, 321082905, 1605444405, 8027354055, 40136925680, 200685295955, 1003427268205, 5017139711105, 25085702537730, 125428529603755, 627142668099880, 3135713425289030, 15678567227192655

Offset: 0

Author

Alois P. Heinz, Sep 08 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=5 of A246935.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 5*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);

Mathematica

(O[x]^20 - 4/QPochhammer[5, x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, 5 b[n-i, i]]]];
a[n_] := b[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

G.f.: Product_{i>=1} 1/(1-5*x^i).

a(n) ~ c * 5^n, where c = Product_{k>=1} 1/(1-1/5^k) = 1.3152135557353452193080... . - Vaclav Kotesovec, Mar 19 2015

G.f.: Sum_{i>=0} 5^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018