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 A074141 Sum of products of parts increased by 1 in all partitions of n. 14
 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 nn, 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 *) 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

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