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A005046 Number of partitions of a 2n-set into even blocks.
(Formerly M3640)
38
1, 1, 4, 31, 379, 6556, 150349, 4373461, 156297964, 6698486371, 337789490599, 19738202807236, 1319703681935929, 99896787342523081, 8484301665702298804, 802221679220975886631, 83877585692383961052499, 9640193854278691671399436, 1211499609050804749310115589 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Louis Comtet, Analyse Combinatoire Tome II, pages 61-62.

L. 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).

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 51 terms from T. D. Noe)

C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.

Steven R. Finch, Moments of sums, April 23, 2004 [Cached copy, with permission of the author]

J. Riordan, Letter, Jul 06 1978

J. Shallit, Letter to N. J. A. Sloane, Jan 13 1976.

Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

FORMULA

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

MAPLE

a:= proc(n) option remember;

`if`(n=0, 1, add(binomial(2*n-1, 2*k-1) *a(n-k), k=1..n))

end:

seq(a(n), n=0..30); # Alois P. Heinz, Apr 12 2011

# 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):

seq(a(n), n=0..18); # after Emanuele Munarini, Peter Luschny, Sep 10 2017

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

MATHEMATICA

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 *)

PROG

(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))

print([a(n) for n in range(21)]) # Indranil Ghosh, Sep 11 2017, after Maple program by Alois P. Heinz

CROSSREFS

See A156289 for the table of partitions of a 2n-set into k even blocks.

For partitions into odd blocks see A003724 and A136630.

Cf. A000110, A003724.

Sequence in context: A145087 A215529 A351798 * A323568 A174324 A211194

Adjacent sequences: A005043 A005044 A005045 * A005047 A005048 A005049

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 5 05:43 EST 2022. Contains 358578 sequences. (Running on oeis4.)