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A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

%I A323433 #27 May 12 2023 18:29:13
%S A323433 1,1,3,5,10,14,25,34,54,74,109,146,211,276,381,501,675,871,1156,1477,
%T A323433 1926,2447,3142,3957,5038,6291,7918,9839,12277,15148,18773,23027,
%U A323433 28333,34587,42284,51357,62466,75503,91344,109971,132421,158755,190365,227354,271511
%N A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.
%H A323433 Alois P. Heinz, <a href="/A323433/b323433.txt">Table of n, a(n) for n = 0..5000</a>
%F A323433 a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
%F A323433 From _Seiichi Manyama_, Jan 21 2022: (Start)
%F A323433 G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
%F A323433 G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
%F A323433 a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - _Ridouane Oudra_, Apr 13 2023
%e A323433 The a(5) = 14 split partitions:
%e A323433   [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
%e A323433 .
%e A323433   [4] [3] [2 1]
%e A323433   [1] [2] [1 1]
%e A323433 .
%e A323433   [3] [2]
%e A323433   [1] [2]
%e A323433   [1] [1]
%e A323433 .
%e A323433   [2]
%e A323433   [1]
%e A323433   [1]
%e A323433   [1]
%e A323433 .
%e A323433   [1]
%e A323433   [1]
%e A323433   [1]
%e A323433   [1]
%e A323433   [1]
%p A323433 b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
%p A323433       [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
%p A323433     end:
%p A323433 a:= n-> `if`(n=0, 1, b(n$2, 0)):
%p A323433 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 15 2019
%t A323433 Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
%t A323433 (* Second program: *)
%t A323433 b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
%t A323433      DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
%t A323433 a[n_] := If[n == 0, 1, b[n, n, 0]];
%t A323433 a /@ Range[0, 50] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%o A323433 (PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ _Seiichi Manyama_, Jan 21 2022
%o A323433 (PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ _Seiichi Manyama_, Jan 21 2022
%Y A323433 Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.
%K A323433 nonn
%O A323433 0,3
%A A323433 _Gus Wiseman_, Jan 15 2019