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A275423
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Number of set partitions of [n] such that five is a multiple of each block size.
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6
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1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 379, 1849, 9109, 37324, 128129, 507508, 3031393, 19609773, 108440893, 500515633, 2467616641, 17154715726, 134519207131, 927764339426, 5359830269641, 31580724696907, 248587878630807, 2259650025239257, 18541914182165557
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OFFSET
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0,6
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LINKS
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FORMULA
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E.g.f.: exp(x+x^5/5!).
a(n) = n! * Sum_{k=0..floor(n/5)} (1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) + binomial(n-1,4) * a(n-5) for n > 4. (End)
a(n) = hypergeom([-n/5,(1-n)/5,(2-n)/5,(3-n)/5,(4-n)/5],[],-625/24))). - Karol A. Penson, Sep 14 2023.
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EXAMPLE
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a(6) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 5]))
end:
seq(a(n), n=0..30);
# second Maple program:
seq(simplify(hypergeom([-n/5, (1-n)/5, (2-n)/5, (3-n)/5, (4-n)/5], [], -625/24)), n = 0..28); # Karol A. Penson, Sep 14 2023.
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 5}}]];
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PROG
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(PARI) a(n) = n!*sum(k=0, n\5, 1/5!^k*binomial(n-4*k, k)/(n-4*k)!); \\ Seiichi Manyama, Feb 26 2022
(PARI) a(n) = if(n<5, 1, a(n-1)+binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 26 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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