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A054440
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Number of ordered pairs of partitions of n with no common parts.
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18
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1, 0, 2, 4, 12, 16, 48, 60, 148, 220, 438, 618, 1302, 1740, 3216, 4788, 8170, 11512, 19862, 27570, 45448, 64600, 100808, 141724, 223080, 307512, 465736, 652518, 968180, 1334030, 1972164, 2691132, 3902432, 5347176, 7611484, 10358426, 14697028, 19790508, 27691500
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OFFSET
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0,3
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LINKS
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Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
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FORMULA
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G.f.: Sum[p(n)^2*x^n]/Sum[p(n)*x^n], with p(n)=number of partitions of n.
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n)) / (64 * 2^(1/4) * n^(7/4)). - Vaclav Kotesovec, May 20 2018
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EXAMPLE
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a(3)=4 because of the 4 pairs of partitions of 3: (3,21),(3,111),(21,3),(111,3).
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MAPLE
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with(combinat): p1 := sum(numbpart(n)^2*x^n, n=0..500): it := p1*product((1-x^i), i=1..500): s := series(it, x, 500): for i from 0 to 100 do printf(`%d, `, coeff(s, x, i)) od:
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^2*x^k, {k, 0, nmax}]/Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
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PROG
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(Haskell)
a054440 = sum . zipWith (*) a087960_list . map a001255 . a260672_row
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Herbert S. Wilf, May 13 2000
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EXTENSIONS
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STATUS
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approved
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