cp's OEIS Frontend

A134737 Number of partitions of the n-th partition number into positive parts not greater than n.

%I A134737 #28 Feb 16 2025 08:33:07
%S A134737 1,2,3,6,13,44,131,638,3060,22367,167672,2127747,26391031,537973241,
%T A134737 12274276512,429819314124,16928838590640,1068323095351171,
%U A134737 75345432929798690,8339062208354516217,1083103359596125913021,209256696715820656730807,48414226122932084106352434
%N A134737 Number of partitions of the n-th partition number into positive parts not greater than n.
%H A134737 Alois P. Heinz, <a href="/A134737/b134737.txt">Table of n, a(n) for n = 1..55</a>
%H A134737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Partition.html">Partition</a>
%H A134737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>
%H A134737 <a href="/index/Par#partN">Index entries for sequences related to partitions</a>
%F A134737 a(n) = A026820(A026820(n,n),n) = A026820(A000041(n),n).
%p A134737 with(numtheory): P:= proc(n) local d, j; P(n):= `if`(n=0, 1, add(add(d, d=divisors(j)) *P(n-j), j=1..n)/n) end: b:= proc(n,i) if n<0 then 0 elif n=0 then 1 elif i=0 then 0 else b(n,i):= b(n, i-1) +b(n-i, i) fi end: a:= n-> b(P(n),n): seq(a(n), n=1..25); # _Alois P. Heinz_, Jul 17 2009
%t A134737 (* first do *) Needs["DiscreteMath`IntegerPartitions`"] (* then *) a[n_] := Length@ IntegerPartitions[ PartitionsP[n], n] (* _Robert G. Wilson v_, Nov 11 2007 *)
%t A134737 P[n_] := P[n] = Module[{d, j}, If[n == 0, 1, Sum[DivisorSum[j, #&]*P[n - j], {j, 1, n}]/n]]; b [n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i == 0, 0, True, b[n, i] = b[n, i-1] + b[n-i, i]]; a[n_] := b[P[n], n]; Table [a[n], {n, 1, 25}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%K A134737 nonn
%O A134737 1,2
%A A134737 _Reinhard Zumkeller_, Nov 07 2007
%E A134737 More terms from _Alois P. Heinz_, Jul 17 2009