A323766 - OEIS

%I #13 Jan 28 2019 07:37:49

%S 1,1,4,5,12,9,25,17,42,39,64,58,132,103,173,200,303,299,491,492,756,

%T 832,1122,1257,1858,1975,2646,3083,4057,4567,6118,6844,8913,10265,

%U 12912,14931,19089,21639,27003,31397,38830,44585,55138,63263,77371,89585,108076

%N Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.

%C Also the number of constant multiset partitions of constant multiset partitions of integer partitions of n.

%H Vaclav Kotesovec, <a href="/A323766/b323766.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - _Vaclav Kotesovec_, Jan 28 2019

%e The a(6) = 25 constant multiset partitions of constant multiset partitions of integer partitions of 6:

%e   ((6))

%e   ((52))

%e   ((42))

%e   ((33))

%e   ((3)(3))

%e   ((3))((3))

%e   ((411))

%e   ((321))

%e   ((222))

%e   ((2)(2)(2))

%e   ((2))((2))((2))

%e   ((3111))

%e   ((2211))

%e   ((21)(21))

%e   ((21))((21))

%e   ((21111))

%e   ((111111))

%e   ((111)(111))

%e   ((11)(11)(11))

%e   ((111))((111))

%e   ((11))((11))((11))

%e   ((1)(1)(1)(1)(1)(1))

%e   ((1)(1)(1))((1)(1)(1))

%e   ((1)(1))((1)(1))((1)(1))

%e   ((1))((1))((1))((1))((1))((1))

%t Table[If[n==0,1,Sum[PartitionsP[d]*DivisorSigma[0,n/d],{d,Divisors[n]}]],{n,0,30}]

%o (PARI) a(n) = if (n==0, 1, sumdiv(n, d, numbpart(d)*numdiv(n/d))); \\ _Michel Marcus_, Jan 28 2019

%Y Cf. A000005, A000041, A000837, A001970, A034729, A047968, A306017, A319066, A323764, A323765, A323774.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 27 2019