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A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

%I A323434 #11 May 18 2021 06:31:28
%S A323434 1,1,1,3,3,5,7,9,11,15,20,24,31,38,48,59,72,86,106,125,150,180,213,
%T A323434 250,296,347,407,477,555,645,751,869,1003,1161,1334,1534,1763,2018,
%U A323434 2306,2637,3002,3418,3886,4409,4994,5659,6390,7214,8135,9160,10300,11580,12990
%N A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.
%H A323434 Alois P. Heinz, <a href="/A323434/b323434.txt">Table of n, a(n) for n = 0..7000</a>
%F A323434 a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.
%e A323434 The a(10) = 20 split partitions:
%e A323434   [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
%e A323434 .
%e A323434   [9] [8] [7] [6] [4 3]
%e A323434   [1] [2] [3] [4] [2 1]
%e A323434 .
%e A323434   [7] [6] [5] [5]
%e A323434   [2] [3] [4] [3]
%e A323434   [1] [1] [1] [2]
%e A323434 .
%e A323434   [4]
%e A323434   [3]
%e A323434   [2]
%e A323434   [1]
%p A323434 b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
%p A323434       `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
%p A323434          b(n-i, min(n-i, i-1), t+1)))
%p A323434     end:
%p A323434 a:= n-> `if`(n=0, 1, b(n$2, 0)):
%p A323434 seq(a(n), n=0..60);  # _Alois P. Heinz_, Jan 15 2019
%t A323434 Table[Sum[Length[Divisors[Length[ptn]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
%t A323434 (* Second program: *)
%t A323434 b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
%t A323434      If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
%t A323434      b[n-i, Min[n-i, i-1], t+1]]];
%t A323434 a[n_] := If[n == 0, 1, b[n, n, 0]];
%t A323434 a /@ Range[0, 60] (* _Jean-François Alcover_, May 18 2021, after _Alois P. Heinz_ *)
%Y A323434 Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.
%K A323434 nonn
%O A323434 0,4
%A A323434 _Gus Wiseman_, Jan 15 2019