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A074141
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Sum of products of parts increased by 1 in all partitions of n.
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11
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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; internal format)
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OFFSET
| 0,2
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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
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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).
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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.
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MATHEMATICA
| Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* T. D. Noe, Nov 01 2011 *)
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CROSSREFS
| Cf. A036035, A074139, A074140, A006906.
Sequence in context: A192873 A017925 A030236 * A122931 A094976 A006869
Adjacent sequences: A074138 A074139 A074140 * A074142 A074143 A074144
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 28 2002
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EXTENSIONS
| More terms from Alford Arnold (Alford1940(AT)aol.com), Sep 17 2002. More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic (vladeta(AT)eunet.rs), Nov 28 2002
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