A210672 -id:A210672 - cp's OEIS frontend

A255929 Expansion of exp( Sum_{n >= 1} A210672(n)*x^n/n ).

1, 2, 15, 308, 13399, 1019106, 119698377, 20039968920, 4527610159068, 1326616296092984, 489092182592254708, 221537815033845709776, 120928125204565597029220, 78286897353506845258973144, 59305342759674536454338570652, 51970719684035315747385128783808

Offset: 0

Peter Bala, Mar 11 2015

It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + 2*x + 15*x^2 + 308*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x)) ) is the o.g.f. for A210672.
This sequence is the particular case m = 2 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k)*u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) has integer coefficients.
For cases see A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1) and A255930(m = 3).
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.

A210672, A241171, A255926(m = -3), A255882(m = -2), A255881(m = -1), A255928(m = 1), A255930(m = 3).

#A255929
A210672 := proc (n) option remember; if n = 0 then 1 else 2*add(binomial(2*n, 2*k)*A210672(k), k = 0 .. n-1) end if; end proc:
A255929 := proc (n) option remember; if n = 0 then 1 else add(A210672(n-k)*A255929(k), k = 0 .. n-1)/n end if; end proc:
seq(A255929(n), n = 0 .. 15);

O.g.f.: exp(2*x + 26*x^2/2 + 842*x^3/3 + 50906*x^4/4 + ...) = 1 + 2*x + 15*x^2 + 308*x^3 + 13399*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A210672(n-k)*a(k) for n >= 1.

A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.

Original entry on oeis.org

1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145, -2404879675441, 370371188237525, -69348874393137901, 15514534163557086905, -4087072509293123892361, 1252259641403629865468285, -441543893249023104553682821, 177519391579539289436664789665

Offset: 0

N. J. A. Sloane

The Euler numbers A000364 with alternating signs.
The first column of the inverse to the matrix with entries C(2*i,2*j), i,j >=0. The full matrix is lower triangular with the i-th subdiagonal having entries a(i)*C(2*j,2*i) j>=i. - Nolan Wallach (nwallach(AT)ucsd.edu), Dec 26 2005
This sequence is also EulerE(2*n). - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
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.
To avoid possible confusion: these are the odd e.g.f. coefficients of Gudermannian(x) with the offset shifted by -1 (even coefficients are zero). They are identical to the even e.g.f. coefficients for 1/cosh(x) = -Gudermannian'(x) (see the Example). Since the complex root of cosh(z) with the smallest absolute value is z0 = i*Pi/2, the radius of convergence for the Taylor series of all these functions is Pi/2 = A019669. - Stanislav Sykora, Oct 07 2016

			Gudermannian(x) = x - (1/6)*x^3 + (1/24)*x^5 - (61/5040)*x^7 + (277/72576)*x^9 + ....
Gudermannian'(x) = 1/cosh(x) = (1/1!)*x^0 - (1/2!)*x^2 + (5/4!)*x^4 - (61/6!)*x^6 + (1385/8!)*x^8 + .... - _Stanislav Sykora_, Oct 07 2016

Gradshteyn and Ryzhik, Tables, 5th ed., Section 1.490, pp. 51-52.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B45.

T. D. Noe, Table of n, a(n) for n = 0..100
Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021.
Filippo Callegaro and Giovanni Gaiffi, On models of the braid arrangement and their hidden symmetries, arXiv preprint arXiv:1406.1304 [math.AT], 2014.
Karl Dilcher and Christophe Vignat, Euler and the Strong Law of Small Numbers, Amer. Math. Mnthly, 123 (May 2016), 486-490.
Allan L. Edmonds and Steven Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
Guodong Liu, On congruences of Euler numbers modulo powers of two, Journal of Number Theory, Volume 128, Issue 12, December 2008, Pages 3063-3071.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 25.
Joe Santmyer, Derivative Polynomials for Trigonometric and Hyperbolic Functions, Arhimede Math. J. (2023) Vol. 10, No. 2, 152-158.
Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 4.
Zhi-Hong Sun, On the further properties of {U_n}, arXiv:1203.5977v1 [math.NT], Mar 27 2012.

Absolute values are the Euler numbers A000364.

Cf. A019669.

A028296 := proc(n) a :=0 ; for k from 1 to 2*n+1 by 2 do a := a+(-1)^((k-1)/2)/2^k/k *add( (-1)^i *(k-2*i)^(2*n+1) *binomial(k,i), i=0..k) ; end do: a ; end proc:
seq(A028296(n),n=0..10) ; # R. J. Mathar, Apr 20 2011

Table[EulerE[2*n], {n, 0, 30}] (* Paul Abbott, Apr 14 2006 *)
Table[(CoefficientList[Series[1/Cosh[x],{x,0,40}],x]*Range[0,40]!)[[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Aug 04 2014*)
With[{nn=40},Take[CoefficientList[Series[Gudermannian[x],{x,0,nn}],x] Range[ 0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Feb 24 2018 *)
{1, Table[2*(-I)*PolyLog[-2*n, I], {n, 1, 12}]} // Flatten (* Peter Luschny, Aug 12 2021 *)
a[0] := 1; a[n_] := a[n] = -Sum[Binomial[2 n, 2 k] a[k], {k, 0, n - 1}]; Map[a, Range[0, 16]] (* Oliver Seipel, May 19 2024 *)

a(n):=sum((-1+(-1)^(k))*(-1)^((k+1)/2)/(2^(k+1)*k)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k),k,1,n); /* with interpolated zeros, Vladimir Kruchinin, Apr 20 2011 */

a(n) = 2*imag(polylog(-2*n, I)); \\ Michel Marcus, May 30 2018

a(n)=eulerfrac(2*n) \\ Charles R Greathouse IV, Mar 23 2022

from sympy import euler
def A028296(n): return euler(n<<1) # Chai Wah Wu, Apr 16 2023

def A028296_list(len):
    f = lambda k: x*(k+1)^2
    g = 1
    for k in range(len-2,-1,-1):
        g = (1-f(k)/(f(k)+1/g)).simplify_rational()
    return taylor(g, x, 0, len-1).list()
print(A028296_list(17))

def A028296(n):
    shapes = ([x*2 for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
print([A028296(n) for n in (0..16)]) # Peter Luschny, Aug 10 2015

E.g.f.: sech(x) = 1/cosh(x) (even terms), or Gudermannian(x) (odd terms).
Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*binomial(2*n, 2*i). - Ralf Stephan, Feb 24 2005
a(n) = Sum_{k=1,3,5,..,2n+1} ((-1)^((k-1)/2) /(2^k*k)) * Sum_{i=0..k} (-1)^i*(k-2*i)^(2n+1) * binomial(k,i). - Vladimir Kruchinin, Apr 20 2011
a(n) = 2^(4*n+1)*(zeta(-2*n,1/4) - zeta(-2*n,3/4)). - Gerry Martens, May 27 2011
From Sergei N. Gladkovskii, Dec 15 2011 - Oct 09 2013: (Start)
Continued fractions:
G.f.: A(x) = 1 - x/(S(0)+x), S(k) = euler(2*k) + x*euler(2*k+2) - x*euler(2*k)* euler(2*k+4)/S(k+1).
E.g.f.: E(x) = 1 - x/(S(0)+x); S(k) = (k+1)*euler(2*k) + x*euler(2*k+2) - x*(k+1)* euler(2*k)*euler(2*k+4)/S(k+1).
2*arctan(exp(z)) - Pi/2 = z*(1 - z^2/(G(0) + z^2)), G(k) = 2*(k+1)*(2*k+3)*euler(2*k) + z^2*euler(2*k+2) - 2*z^2*(k+1)*(2*k+3)*euler(2*k)*euler(2*k+4)/G(k+1).
G.f.: A(x) = 1/S(0) where S(k) = 1 + x*(k+1)^2/S(k+1).
G.f.: 1/Q(0) where Q(k) = 1 - x*k*(3*k-1) - x*(k+1)*(2*k+1)*(x*k^2-1)/Q(k+1).
E.g.f.:(2 - x^4/( (x^2+2)*Q(0) + 2))/(2+x^2) where Q(k) = 4*k + 4 + 1/( 1 - x^2/( 2 + x^2 + (2*k+3)*(2*k+4)/Q(k+1))).
E.g.f.: 1/cosh(x) = 8*(1-x^2)/(8 - 4*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)));
G.f.: 1/U(0) where U(k) = 1 - x + x*(2*k+1)*(2*k+2)/(1 + x*(2*k+1)*(2*k+2)/U(k+1));
G.f.: 1 - x/G(0) where G(k) = 1 - x + x*(2*k+2)*(2*k+3)/(1 + x*(2*k+2)*(2*k+3)/G(k+1));
G.f.: 1/Q(0), where Q(k) = 1 - sqrt(x) + sqrt(x)*(k+1)/(1-sqrt(x)*(k+1)/Q(k+1));
G.f.: (1/Q(0) + 1)/(1-sqrt(x)), where Q(k) = 1 - 1/sqrt(x) + (k+1)*(k+2)/Q(k+1);
G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 + 1/Q(k+1)). (End)
a(n) ~ (-1)^n * (2*n)! * 2^(2*n+2) / Pi^(2*n+1). - Vaclav Kotesovec, Aug 04 2014
a(n) = 2*Im(Li_{-2n}(i)), where Li_n(x) is polylogarithm, i=sqrt(-1). - Vladimir Reshetnikov, Oct 22 2015

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

Original entry on oeis.org

1, 1, 7, 121, 3907, 202741, 15430207, 1619195761, 224061282907, 39531606447181, 8661323866026007, 2307185279184885001, 734307168916191403507, 275199311597682485597221, 119956934012963778952439407

Offset: 0

Ralf Stephan, Apr 30 2004

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

Seiichi Manyama, Table of n, a(n) for n = 0..235 (terms 0..40 from Peter Luschny)
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.

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

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

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-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)

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 */

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 */

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

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

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

Corrected definition, Joerg Arndt, Apr 26 2011

A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -2, 22, -602, 30742, -2523002, 303692662, -50402079002, 11030684333782, -3077986048956602, 1066578948824962102, -449342758735568563802, 226182806795367665865622, -134065091768709178087428602, 92423044260377387363207812342, -73323347841467639992211297199002

Offset: 0

N. J. A. Sloane, Mar 28 2012

sign

The version without signs has an interpretation as a sum over marked Schröder paths. See the Josuat-Verges and Kim reference.
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.
Apparently a(n) = 2*(-1)^n*A002114(n). - R. J. Mathar, Mar 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200
Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.
Zhi-Hong Sun, On the further properties of U_n, arXiv:1203.5977 [math.NT], 2012.

Cf. A002114, A028296, A094088, A210657, A210672, A210674, A210676, A249939, A255882, A241171.

A210657:=proc(n) option remember;
   if n=0 then 1
   else -2*add(binomial(2*n,2*k)*procname(n-k),k=1..floor(n)); fi;
end;
[seq(f(n),n=0..20)];
# Second program:
a := (n) -> 2*36^n*(Zeta(0,-n*2,1/6)-Zeta(0,-n*2,2/3)):
seq(a(n), n=0..15); # Peter Luschny, Mar 11 2015

nmax=20; Table[(CoefficientList[Series[1/(2*Cosh[x]-1), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)
Table[9^n EulerE[2 n, 1/3], {n, 0, 20}] (* Vladimir Reshetnikov, Jun 05 2016 *)

a(n)=polcoeff(sum(m=0, n, (2*m)!*(-x)^m/prod(k=1, m, 1-k^2*x +x*O(x^n)) ), n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 17 2012

O.g.f.: Sum_{n>=0} (2*n)! * (-x)^n / Product_{k=1..n} (1 - k^2*x). - Paul D. Hanna, Sep 17 2012
E.g.f.: 1/(2*cosh(x) - 1) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. - Paul D. Hanna, Oct 30 2014
E.g.f.: cos(z/2)/cos(3z/2) = Sum_{n>=0} abs(a(n))*x^(2*n)/(2*n)!. - Olivier Gérard, Feb 12 2014
From Peter Bala, Mar 09 2015: (Start)
a(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial.
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(3*k + 1)^2).
O.g.f. as a continued fraction: 1/(1 + (3^2 - 1^2)*x/(4 + 12^2*x/(4 + (18^2 - 2^2)*x/(4 + 24^2*x/(4 + (30^2 - 2^2)*x/(4 + 36^2*x/(4 + ... ))))))) = 1 - 2*x + 22*x^2 - 602*x^3 + 30742*x^4 - .... See Josuat-Vergès and Kim, p. 23.
The expansion of exp( Sum_{n >= 1} a(n)*x^n/n ) appears to have integer coefficients. See A255882. (End)
a(n) = 2*36^n*(zeta(-n*2,1/6)-zeta(-n*2,2/3)), where zeta(a,z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) ~ 2 * (-1)^n * (2*n)! * 3^(2*n+1/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 14 2015
a(n) = Sum_{k=0..n} A241171(n, k)*(-2)^k. - Peter Luschny, Sep 03 2022

A249939 E.g.f.: 1/(5 - 4*cosh(x)).

Original entry on oeis.org

1, 4, 100, 6244, 727780, 136330084, 37455423460, 14188457293924, 7087539575975140, 4514046217675793764, 3570250394992512270820, 3433125893070920512725604, 3944372161432193963534198500, 5336301013125557989981503385444, 8396749419933421378024498580446180

Offset: 0

Paul D. Hanna, Nov 19 2014

nonn

a(n) = 4*A242858(2*n) for n>0.
a(n) = A249940(n)/3.
a(n) == 4 (mod 96) for n>0.

			E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
where 3*A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 + 2*exp(49*x)/2^7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A249938, A249940, A247082, A250914, A250915, A242858.

Cf. A210676, A210657, A028296, A094088, A210672, A210674.

/* E.g.f.: 1/(5 - 4*cosh(x)) */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.

A210676 a(0)=1; thereafter a(n) = -3Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, -3, 51, -2163, 171231, -21785223, 4065116811, -1045879150683, 354837765112791, -153492920593758543, 82453488412268175171, -53850296379425229208803, 42020794900180632536559951, -38611325264740403135096141463, 41264215393801752999038147563131, -50749285521783354479522581233836523

Offset: 0

N. J. A. Sloane, Mar 28 2012

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.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255926. - Peter Bala, Mar 13 2015
In general, for c<>0 is e.g.f. = 1/(c+1-c*cosh(x)) (even coefficients). For c > 0 is a(n) ~ 2 * (2*n)! / (sqrt(2*c+1) * (arccosh((c+1)/c))^(2*n+1)). For c < 0 is a(n) ~ (-1)^n * (2*n)! / (sqrt(-2*c-1) * 2^(2*n) * arccos(sqrt((2*c + 1) / (2*c)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A028296, A094088, A210657, A210672, A210674, A249939, A255926.

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(-3);

nmax=20; Table[(CoefficientList[Series[1/(3*Cosh[x]-2), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

E.g.f.: 1/(3*cosh(x)-2) (even coefficients). - Vaclav Kotesovec, Mar 14 2015

a(n) ~ (-1)^n * (2*n)! / (sqrt(5) * 2^(2*n) * (arccos(sqrt(5/6)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

A210674 a(0)=1; thereafter a(n) = 3Sum_{k=1..n} binomial(2n,2k)a(n-k).

Original entry on oeis.org

1, 3, 57, 2703, 239277, 34041603, 7103141697, 2043564786903, 775293596155317, 375019773885750603, 225270492555606688137, 164517775480287009524703, 143555042043378357951428157, 147502150365016885913874781203, 176273363579960990244526939543377, 242422256082395157286909073370272103

Offset: 0

N. J. A. Sloane, Mar 28 2012

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.
Exp( Sum_{n >= 1} a(n)*x^n/n) is the o.g.f. for A255930. - Peter Bala, Mar 13 2015
In general, for c > 0 is a(n) ~ sqrt(Pi/(2*c+1)) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((c + 1 + sqrt(2*c+1)) / c))^(2*n+1)) = 2*(2*n)!/(sqrt(2*c+1)*(arccosh((c+1)/c))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015
For c < 0 is a(n) ~ (-1)^n * (2*n)! / (sqrt(-2*c-1) * 2^(2*n) * arccos(sqrt((2*c + 1)/(2*c)))^(2*n+1)). - Vaclav Kotesovec, Mar 14 2015

G. C. Greubel, Table of n, a(n) for n = 0..210

Cf. A210676 (c=-3), A210657 (c=-2), A028296 (c=-1), A094088 (c=1), A210672 (c=2), A249939 (c=4).

Cf. A255930.

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(3);

nmax=20; Table[(CoefficientList[Series[1/(4-3*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*n+1]], {n,0,nmax}] (* Vaclav Kotesovec, Mar 14 2015 *)

a(n) ~ sqrt(Pi/7) * 2^(2*n+2) * n^(2*n+1/2) / (exp(2*n) * (log((4 + sqrt(7)) / 3))^(2*n+1)). - Vaclav Kotesovec, Mar 13 2015

E.g.f.: 1/(4-3*cosh(x)) (even coefficients). - Vaclav Kotesovec, Mar 14 2015

cp's OEIS Frontend

A255929 Expansion of exp( Sum_{n >= 1} A210672(n)*x^n/n ).

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A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.

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

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A210657 a(0)=1; thereafter a(n) = -2*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

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A249939 E.g.f.: 1/(5 - 4*cosh(x)).

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A210676 a(0)=1; thereafter a(n) = -3*Sum_{k=1..n} binomial(2n,2k)*a(n-k).

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A210657 a(0)=1; thereafter a(n) = -2Sum_{k=1..n} binomial(2n,2k)a(n-k).

A210676 a(0)=1; thereafter a(n) = -3Sum_{k=1..n} binomial(2n,2k)a(n-k).

A210674 a(0)=1; thereafter a(n) = 3Sum_{k=1..n} binomial(2n,2k)a(n-k).