cp's OEIS Frontend

A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).

%I A094088 #74 Jan 16 2022 11:09:21
%S A094088 1,1,7,121,3907,202741,15430207,1619195761,224061282907,
%T A094088 39531606447181,8661323866026007,2307185279184885001,
%U A094088 734307168916191403507,275199311597682485597221,119956934012963778952439407
%N A094088 E.g.f. 1/(2-cosh(x)) (even coefficients).
%C A094088 With alternating signs, e.g.f.: 1/(2-cos(x)).
%C A094088 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.
%C A094088 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.
%C A094088 a(n) is the number of ordered set partitions of {1,2,...,2n} into even size blocks. - _Geoffrey Critzer_, Dec 03 2012
%C A094088 Except for a(0), row sums of A241171. - _Peter Bala_, Aug 20 2014
%C A094088 Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255928. - _Peter Bala_, Mar 13 2015
%C A094088 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
%H A094088 Seiichi Manyama, <a href="/A094088/b094088.txt">Table of n, a(n) for n = 0..235</a> (terms 0..40 from Peter Luschny)
%H A094088 J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
%F A094088 1/(2-cosh(x)) = Sum_{n>=0} a(n)x^(2n)/(2n)! = 1 + x^2/2 + 7x^4/24 + 121x^6/720 + ...
%F A094088 Recurrence: a(0)=1, a(n) = Sum_{k=1..n} C(2n, 2n-2k)*a(n-k).
%F A094088 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
%F A094088 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
%F A094088 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
%F A094088 a(n) ~ 2*(2*n)! /(sqrt(3) * (log(2+sqrt(3)))^(2*n+1)). - _Vaclav Kotesovec_, Oct 19 2013
%p A094088 f:=proc(n,k) option remember;  local i;
%p A094088 if n=0 then 1
%p A094088 else k*add(binomial(2*n,2*i)*f(n-i,k),i=1..floor(n)); fi; end;
%p A094088 g:=k->[seq(f(n,k),n=0..40)];g(1); # _N. J. A. Sloane_, Mar 28 2012
%t A094088 nn=30;Select[Range[0,nn]!CoefficientList[Series[1/(2-Cosh[x]),{x,0,nn}],x],#>0&]  (* _Geoffrey Critzer_, Dec 03 2012 *)
%t A094088 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-François Alcover_, Apr 03 2015, after _Vladimir Kruchinin_ *)
%o A094088 (PARI) a(n) = if (n == 0, 1, sum(k=1, n, binomial(2*n, 2*n-2*k)*a(n-k)));
%o A094088 (Maxima)
%o A094088 a(n):=b(2*n+2);
%o A094088 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 */
%o A094088 (Sage)
%o A094088 def A094088(n) :
%o A094088     @CachedFunction
%o A094088     def intern(n) :
%o A094088         if n == 0 : return 1
%o A094088         if n % 2 != 0 : return 0
%o A094088         return add(intern(k)*binomial(n,k) for k in range(n)[::2])
%o A094088     return intern(2*n)
%o A094088 [A094088(n) for n in (0..14)]  # _Peter Luschny_, Jul 14 2012
%o A094088 (Maxima)
%o A094088 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 */
%Y A094088 Cf. A241171, A210676, A210657, A028296, A210672, A210674, A255928.
%K A094088 nonn,easy
%O A094088 0,3
%A A094088 _Ralf Stephan_, Apr 30 2004
%E A094088 Corrected definition, _Joerg Arndt_, Apr 26 2011