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A005046
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Number of partitions of a 2n-set into even blocks.
(Formerly M3640)
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40
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1, 1, 4, 31, 379, 6556, 150349, 4373461, 156297964, 6698486371, 337789490599, 19738202807236, 1319703681935929, 99896787342523081, 8484301665702298804, 802221679220975886631, 83877585692383961052499, 9640193854278691671399436, 1211499609050804749310115589
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history;
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OFFSET
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0,3
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COMMENTS
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Conjecture: Taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, 1, 4, 1, 9, 6, 9, ...], with an apparent period [1, 4, 1, 9, 6, 9] beginning at a(1), of length 6. Cf. A006154. - Peter Bala, Apr 12 2023
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REFERENCES
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Louis Comtet, Analyse Combinatoire Tome II, pages 61-62.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 3rd line of table.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Steven R. Finch, Moments of sums, April 23, 2004 [Cached copy, with permission of the author]
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FORMULA
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E.g.f.: exp(cosh(x) - 1) (or exp(cos(x)-1) ).
a(n) = Sum_{k=1..n} binomial(2*n-1, 2*k-1)*a(n-k). - Vladeta Jovovic, Apr 10 2003
a(n) = sum(1/k!*sum(binomial(k,m)/(2^(m-1))*sum(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n), n>0. - Vladimir Kruchinin, Aug 05 2010
a(n) = Sum_{k=1..2*n} Sum_{i=0..k-1} ((i-k)^(2*n)*binomial(2*k,i)*(-1)^i)/(2^(k-1)*k!), n>0, a(0)=1. - Vladimir Kruchinin, Oct 04 2012
E.g.f.: E(0)-1, where E(k) = 2 + (cosh(x)-1)/(2*k + 1 - (cosh(x)-1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 23 2013
a(n) = Sum_{k=0..2*n} binomial(2*n,k)*(-1)^k*S_k(1/2)*S_{2*n-k}( 1/2), where S_n(x) is the n-th Bell polynomial (or exponential polynomial). - Emanuele Munarini, Sep 10 2017
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MAPLE
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a:= proc(n) option remember;
`if`(n=0, 1, add(binomial(2*n-1, 2*k-1) *a(n-k), k=1..n))
end:
# second Maple program:
a := n -> add(binomial(2*n, k)*(-1)^k*BellB(k, 1/2)*BellB(2*n-k, 1/2), k=0..2*n):
B := BellMatrix(n -> modp(n, 2), 31): # defined in A264428.
seq(add(k, k in B[2*n + 1]), n=0..15); # Peter Luschny, Aug 13 2019
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MATHEMATICA
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NestList[ Factor[ D[#, {x, 2}]] &, Exp[ Cosh[x] - 1], 16] /. x -> 0
a[0] = 1; a[n_] := Sum[Sum[(i-k)^(2*n)*Binomial[2*k, i]*(-1)^i, {i, 0, k-1}]/(2^(k-1)*k!), {k, 1, 2*n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 07 2015, after Vladimir Kruchinin *)
Table[Sum[BellY[2 n, k, 1 - Mod[Range[2 n], 2]], {k, 0, 2 n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
With[{nn=40}, Abs[Take[CoefficientList[Series[Exp[Cos[x]-1], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]]] (* Harvey P. Dale, Feb 06 2017 *)
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PROG
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(Maxima) a(n):= sum(1/k!*sum(binomial(k, m)/(2^(m-1))*sum(binomial(m, j) *(2*j-m)^(2*n), j, 0, m/2)*(-1)^(k-m), m, 0, k), k, 1, 2*n); \\ Vladimir Kruchinin, Aug 05 2010
(Maxima) a(n):=sum(sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(i), i, 0, k-1)/(2^(k-1)*k!), k, 1, 2*n); \\ Vladimir Kruchinin, Oct 04 2012
(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n): return 1 if n==0 else sum(binomial(2*n - 1, 2*k - 1)*a(n - k) for k in range(1, n + 1))
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CROSSREFS
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See A156289 for the table of partitions of a 2n-set into k even blocks.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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