OFFSET
0,3
COMMENTS
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
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.
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved