login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A074141 Sum of products of parts increased by 1 in all partitions of n. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 1/Product_{m>0} (1-(m+1)*x^m). Recurrence: 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 = 18.56314656361011472747535423226928404842588594722907068201... . - Vaclav Kotesovec, Sep 11 2014

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, 1, `if`(i<1, 0,

      b(n, i-1) +`if`(i>n, 0, (1+i)*b(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 *)

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

Cf. A036035, A074139, A074140, A006906.

Sequence in context: A192873 A017925 A030236 * A122931 A094976 A006869

Adjacent sequences:  A074138 A074139 A074140 * A074142 A074143 A074144

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 23 14:38 EDT 2014. Contains 248465 sequences.