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Search: id:A024786
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| A024786 |
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Number of 2's in all partitions of n. |
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+0
18
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| 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, 295, 389, 526, 686, 911, 1176, 1538, 1968, 2540, 3223, 4115, 5181, 6551, 8191, 10269, 12756, 15873, 19598, 24222, 29741, 36532, 44624, 54509, 66261, 80524, 97446, 117862, 142029, 171036, 205290, 246211
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Also number of partitions of n-1 with a distinguished partition different from all the others.
In general the number of times that j appears in the partitions of n equals Sum_{k<n, k = n (mod j)} P(k). In particular this gives a formula for a(n), A024787, ..., A024794, for j = 2,...,10; it generalizes the formula given for A000070 for j=1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
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REFERENCES
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J. Riordan, Combinatorial Identities, Wiley, 1968, p. 184.
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FORMULA
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a(n) = sum{k=1 to floor(n/2)} A000041(n-2k).
a(n) = Sum_{k<n, k = n (mod 2)} P(k), P(k) =number of partitions of k as in A000041, P(0) = 1. - Jose Luis Arregui (arregui(AT)posta.unizar.es), Apr 05 2002
G.f.: x/((1-x)*(1-x^2)^2))*product(1/(1-x^j), j=3..infty) from Riordan reference second term, last eq.
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MATHEMATICA
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<< DiscreteMath`Combinatorica`; Table[ Count[ Flatten[ Partitions[n]], 2], {n, 1, 50} ]
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CROSSREFS
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Cf. A066633, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
Column 2 of A060244.
First differences of A000097.
Adjacent sequences: A024783 A024784 A024785 this_sequence A024787 A024788 A024789
Sequence in context: A099108 A001994 A084421 this_sequence A097497 A006167 A137504
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
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Formula and comment from Christian G. Bower (bowerc(AT)usa.net), Jun 22 2000
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