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%I M3640 #130 Nov 14 2023 15:11:23
%S 1,1,4,31,379,6556,150349,4373461,156297964,6698486371,337789490599,
%T 19738202807236,1319703681935929,99896787342523081,
%U 8484301665702298804,802221679220975886631,83877585692383961052499,9640193854278691671399436,1211499609050804749310115589
%N Number of partitions of a 2n-set into even blocks.
%C 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
%D Louis Comtet, Analyse Combinatoire Tome II, pages 61-62.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 3rd line of table.
%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
%D L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
%D L. Lovasz, Combinatorial Problems and Exercises, North-Holland, 1993, pp. 15.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005046/b005046.txt">Table of n, a(n) for n = 0..250</a> (first 51 terms from T. D. Noe)
%H C. Ahmed, P. Martin, and V. Mazorchuk, <a href="http://arxiv.org/abs/1503.06718">On the number of principal ideals in d-tonal partition monoids</a>, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
%H Steven R. Finch, <a href="/A000296/a000296.pdf">Moments of sums</a>, April 23, 2004 [Cached copy, with permission of the author]
%H Vladimir Victorovich Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>
%H J. Shallit, <a href="/A005046/a005046.pdf">Letter to N. J. A. Sloane</a>, Jan 13 1976.
%H Sebastian Volz, <a href="https://www.uni-saarland.de/fileadmin/upload/lehrstuhl/weber-moritz/Abschlussarbeiten/volz-bachelors-thesis.pdf">Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets</a>, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
%F E.g.f.: exp(cosh(x) - 1) (or exp(cos(x)-1) ).
%F a(n) = Sum_{k=1..n} binomial(2*n-1, 2*k-1)*a(n-k). - _Vladeta Jovovic_, Apr 10 2003
%F 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
%F 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
%F 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
%F 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
%p a:= proc(n) option remember;
%p `if`(n=0, 1, add(binomial(2*n-1, 2*k-1) *a(n-k), k=1..n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 12 2011
%p # second Maple program:
%p a := n -> add(binomial(2*n,k)*(-1)^k*BellB(k,1/2)*BellB(2*n-k,1/2), k=0..2*n):
%p seq(a(n), n=0..18); # after _Emanuele Munarini_,_Peter Luschny_, Sep 10 2017
%p B := BellMatrix(n -> modp(n, 2), 31): # defined in A264428.
%p seq(add(k, k in B[2*n + 1]),n=0..15); # _Peter Luschny_, Aug 13 2019
%t NestList[ Factor[ D[#, {x, 2}]] &, Exp[ Cosh[x] - 1], 16] /. x -> 0
%t 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_ *)
%t Table[Sum[BellY[2 n, k, 1 - Mod[Range[2 n], 2]], {k, 0, 2 n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%t 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 *)
%o (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
%o (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
%o (Python)
%o from sympy.core.cache import cacheit
%o from sympy import binomial
%o @cacheit
%o 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))
%o print([a(n) for n in range(21)]) # _Indranil Ghosh_, Sep 11 2017, after Maple program by _Alois P. Heinz_
%Y See A156289 for the table of partitions of a 2n-set into k even blocks.
%Y For partitions into odd blocks see A003724 and A136630.
%Y Cf. A000110, A003724.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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