login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)