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

%I A074141 #52 May 10 2021 04:34:27
%S A074141 1,2,7,18,50,118,301,684,1621,3620,8193,17846,39359,84198,181313,
%T A074141 383208,811546,1695062,3546634,7341288,15207022,31261006,64255264,
%U A074141 131317012,268336125,545858260,1110092387,2250057282,4558875555,9213251118,18613373708,37529713890
%N A074141 Sum of products of parts increased by 1 in all partitions of n.
%C A074141 Replace each term in A036035 by the number of its divisors as in A074139; sequence gives sum of terms in the n-th row.
%C A074141 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
%H A074141 Alois P. Heinz, <a href="/A074141/b074141.txt">Table of n, a(n) for n = 0..3317</a>
%F A074141 G.f.: 1/Product_{m>0} (1-(m+1)*x^m).
%F A074141 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).
%F A074141 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
%F A074141 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
%e A074141 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.
%p A074141 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A074141       2^n, b(n, i-1) +(1+i)*b(n-i, min(n-i, i)))
%p A074141     end:
%p A074141 a:= n-> b(n$2):
%p A074141 seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 07 2014
%t A074141 Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* _T. D. Noe_, Nov 01 2011 *)
%t A074141 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_ *)
%o A074141 (Maxima)
%o A074141 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);
%o A074141 makelist(S(n,1),n,0,17); /* _Vladimir Kruchinin_, Sep 07 2014 */
%Y A074141 Cf. A006906, A036035, A074140, A256155.
%Y A074141 Row sums of A074139 and of A079025 and of A079308 and of A238963.
%Y A074141 Column k=2 of A261718.
%Y A074141 Cf. A267008.
%K A074141 nonn
%O A074141 0,2
%A A074141 _Amarnath Murthy_, Aug 28 2002
%E A074141 More terms from _Alford Arnold_, Sep 17 2002
%E A074141 More terms, better description and formulas from _Vladeta Jovovic_, _Vladimir Baltic_, Nov 28 2002