A074141 Sum of products of parts increased by 1 in all partitions of n.

1, 2, 7, 18, 50, 118, 301, 684, 1621, 3620, 8193, 17846, 39359, 84198, 181313, 383208, 811546, 1695062, 3546634, 7341288, 15207022, 31261006, 64255264, 131317012, 268336125, 545858260, 1110092387, 2250057282, 4558875555, 9213251118, 18613373708, 37529713890

Offset: 0

Author

Amarnath Murthy, Aug 28 2002

Keywords

• nonn

Comments

Replace each term in A036035 by the number of its divisors as in A074139; sequence gives sum of terms in the n-th row.

This is the sum of the number of submultisets of the multisets with n elements; a part of a partition is a frequency of such an element. - George Beck, Nov 01 2011

Examples

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products when parts are increased by 1 are 5,8,9,12,16 and their sum is a(4) = 50.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3317

Crossrefs

Cf. A006906, A036035, A074140, A256155.

Row sums of A074139 and of A079025 and of A079308 and of A238963.

Column k=2 of A261718.

Cf. A267008.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +(1+i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Sep 07 2014

Mathematica

Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* T. D. Noe, Nov 01 2011 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, (1+i) * b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

Maxima

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

Formula

G.f.: 1/Product_{m>0} (1-(m+1)*x^m).

a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*(d+1)^(k/d).

a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k+1)*S(n-k,k))+(n+1), S(n,n)=n+1, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014

a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1-(k+1)/2^k) = 18.56314656361011472747535423226928404842588594722907068201... = A256155. - Vaclav Kotesovec, Sep 11 2014, updated May 10 2021

Extensions

More terms from Alford Arnold, Sep 17 2002

More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic, Nov 28 2002