A279786 - OEIS

%I #16 Jul 17 2018 21:50:50

%S 1,1,2,4,5,9,16,22,28,49,69,94,138,187,257,374,479,639,886,1146,1577,

%T 2103,2676,3534,4620,5910,7542,9816,12650,15986,20538,25740,32632,

%U 41442,51383,64771,81281,100729,125041,155557,192641,236810,293593,359880,441276

%N Twice-partitioned numbers where the first partition is strict and the latter partitions are constant.

%H Alois P. Heinz, <a href="/A279786/b279786.txt">Table of n, a(n) for n = 0..8000</a>

%H Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>

%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*d(j)^k*x^(j*k)/k), where d(j) is the number of the divisors of j (A000005). - _Ilya Gutkovskiy_, Jul 17 2018

%e The a(5)=9 twice-partitions are:

%e ((5)), ((4)(1)), ((3)(2)), ((3)(11)), ((22)(1)),

%e ((111)(2)), ((111)(11)), ((1111)(1)), ((11111)).

%p b:= proc(n, i) option remember;

%p       `if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+

%p       `if`(i>n, 0, numtheory[tau](i)*b(n-i, i-1))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..70);  # _Alois P. Heinz_, Dec 20 2016

%t nn=20;CoefficientList[Series[Product[(1+DivisorSigma[0,n]x^n),{n,nn}],{x,0,nn}],x]

%Y Cf. A000005, A000009, A063834.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 18 2016