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

%I A246937 #16 Jan 02 2021 16:07:11
%S A246937 1,5,30,155,805,4055,20455,102455,513230,2567230,12841130,64211380,
%T A246937 321082905,1605444405,8027354055,40136925680,200685295955,
%U A246937 1003427268205,5017139711105,25085702537730,125428529603755,627142668099880,3135713425289030,15678567227192655
%N A246937 Number of partitions of n into 5 sorts of parts.
%H A246937 Alois P. Heinz, <a href="/A246937/b246937.txt">Table of n, a(n) for n = 0..1000</a>
%F A246937 G.f.: Product_{i>=1} 1/(1-5*x^i).
%F A246937 a(n) ~ c * 5^n, where c = Product_{k>=1} 1/(1-1/5^k) = 1.3152135557353452193080... . - _Vaclav Kotesovec_, Mar 19 2015
%F A246937 G.f.: Sum_{i>=0} 5^i*x^i/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018
%p A246937 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246937       b(n, i-1) +`if`(i>n, 0, 5*b(n-i, i))))
%p A246937     end:
%p A246937 a:= n-> b(n$2):
%p A246937 seq(a(n), n=0..25);
%t A246937 (O[x]^20 - 4/QPochhammer[5, x])[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%t A246937 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]]]];
%t A246937 a[n_] := b[n, n];
%t A246937 a /@ Range[0, 25] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)
%Y A246937 Column k=5 of A246935.
%K A246937 nonn
%O A246937 0,2
%A A246937 _Alois P. Heinz_, Sep 08 2014