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A094088 E.g.f. 1/(2-cosh(x)) (even coefficients). 17
1, 1, 7, 121, 3907, 202741, 15430207, 1619195761, 224061282907, 39531606447181, 8661323866026007, 2307185279184885001, 734307168916191403507, 275199311597682485597221, 119956934012963778952439407 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

With alternating signs, e.g.f.: 1/(2-cos(x)).

7 divides a(3n+2). Ira Gessel remarks: For any odd prime p, the coefficients of 1/(2-cosh(x)) as e.g.f. are periodic with period dividing p-1.

Consider the sequence defined by a(0) = 1; thereafter a(n) = c*Sum_{k = 1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.

a(n) is the number of ordered set partitions of {1,2,...,2n} into even size blocks. - Geoffrey Critzer, Dec 03 2012

Except for a(0), row sums of A241171. - Peter Bala, Aug 20 2014

Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255928. - Peter Bala, Mar 13 2015

Also the 2-packed words of degree n; cf. A011782, A000670, A094088, A243664, A243665, A243666 for k-packed words for 0<=k<=5. - Peter Luschny, Jul 06 2015

LINKS

Peter Luschny, Table of n, a(n) for n = 0..40

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

FORMULA

1/(2-cosh(x)) = sum[n>=0, a(n)x^(2n)/(2n)! ] = 1 + x^2/2 + 7x^4/24 + 121x^6/720 + ...

Recurrence: a(0)=1, a(n) = sum[k=1..n, C(2n, 2n-2k)*a(n-k)].

a(0)=1 and, for n>=1, a(n)=b(2*n) where b(n) = sum(k=1..n/2,((sum(j=1..k, ((sum(i=0..j,(j-2*i)^n*binomial(j,i)))*(-1)^(k-j)*binomial(k,j))/2^(j)))*((-1)^n+1))/2). - Vladimir Kruchinin, Apr 23 2011

E.g.f.: 1/(2-cosh(x))=8*(1-x^2)/(8 - 12*x^2 + x^4*U(0))  where U(k)= 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 + 8*(k+1)*(k+2)*(k+3)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 30 2012

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))), n>0, a(0)=1. - Vladimir Kruchinin, Oct 05 2012

a(n) ~ 2*(2*n)! /(sqrt(3) * (log(2+sqrt(3)))^(2*n+1)). - Vaclav Kotesovec, Oct 19 2013

MAPLE

f:=proc(n, k) option remember;  local i;

if n=0 then 1

else k*add(binomial(2*n, 2*i)*f(n-i, k), i=1..floor(n)); fi; end;

g:=k->[seq(f(n, k), n=0..40)]; g(1); # N. J. A. Sloane, Mar 28 2012

MATHEMATICA

nn=30; Select[Range[0, nn]!CoefficientList[Series[1/(2-Cosh[x]), {x, 0, nn}], x], #>0&]  (* Geoffrey Critzer, Dec 03 2012 *)

a[0]=1; a[n_] := Sum[1/2*(1+(-1)^(2*n))*Sum[((-1)^(k-j)*Binomial[k, j]*Sum[(j-2*i )^(2*n)*Binomial[j, i], {i, 0, j}])/2^j, {j, 1, k}], {k, 1, n}]; Table[a[n], {n, 0, 14}] (* Jean-Fran├žois Alcover, Apr 03 2015, after Vladimir Kruchinin *)

PROG

(PARI) a(n) = if (n == 0, 1, sum(k=1, n, binomial(2*n, 2*n-2*k)*a(n-k)));

(Maxima)

a(n):=b(2*n+2);

b(n):=sum(((sum(((sum((j-2*i)^n*binomial(j, i), i, 0, j))*(-1)^(k-j)*binomial(k, j))/2^(j), j, 1, k))*((-1)^n+1))/2, k, 1, n/2); /* Vladimir Kruchinin, Apr 23 2011 */

(Sage)

def A094088(n) :

    @CachedFunction

    def intern(n) :

        if n == 0 : return 1

        if n % 2 != 0 : return 0

        return add(intern(k)*binomial(n, k) for k in range(n)[::2])

    return intern(2*n)

[A094088(n) for n in (0..14)]  # Peter Luschny, Jul 14 2012

(Maxima)

a(n):=sum(sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(i), i, 0, k-1)/(2^(k-1)), k, 1, 2*n); /* Vladimir Kruchinin, Oct 05 2012 */

CROSSREFS

Cf. A241171, A210676, A210657, A028296, A210672, A210674, A255928.

Sequence in context: A274267 A012028 A279235 * A012043 A316730 A211103

Adjacent sequences:  A094085 A094086 A094087 * A094089 A094090 A094091

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Apr 30 2004

EXTENSIONS

Corrected definition, Joerg Arndt, Apr 26 2011

STATUS

approved

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Last modified September 27 22:36 EDT 2020. Contains 337388 sequences. (Running on oeis4.)