A005046 Number of partitions of a 2n-set into even blocks.
1, 1, 4, 31, 379, 6556, 150349, 4373461, 156297964, 6698486371, 337789490599, 19738202807236, 1319703681935929, 99896787342523081, 8484301665702298804, 802221679220975886631, 83877585692383961052499, 9640193854278691671399436, 1211499609050804749310115589
Offset: 0
References
- 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).
Links
- 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]
- Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
- J. Riordan, Letter, Jul 06 1978
- J. Shallit, Letter to N. J. A. Sloane, Jan 13 1976.
- Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
Crossrefs
Programs
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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 *) -
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 */
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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 */
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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
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
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