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A074141
Sum of products of parts increased by 1 in all partitions of n.
20
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
0,2
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
LINKS
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 n<m. - Vladimir 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
EXAMPLE
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.
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 *)
PROG
(Maxima)
S(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then n+1 else sum((k+1)*S(n-k, k), k, m, n/2)+(n+1);
makelist(S(n, 1), n, 0, 17); /* Vladimir Kruchinin, Sep 07 2014 */
CROSSREFS
Row sums of A074139 and of A079025 and of A079308 and of A238963.
Column k=2 of A261718.
Cf. A267008.
Sequence in context: A192873 A017925 A030236 * A122931 A094976 A280598
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Aug 28 2002
EXTENSIONS
More terms from Alford Arnold, Sep 17 2002
More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic, Nov 28 2002
STATUS
approved