A188514 -id:A188514 - cp's OEIS frontend

A255881 Expansion of exp( Sum_{n >= 1} A000364(n)*x^n/n ).

1, 1, 3, 23, 371, 10515, 461869, 28969177, 2454072147, 269732425859, 37312477130105, 6342352991066661, 1299300852841580893, 315702973949640373933, 89765549161833322593411, 29526682496433138896248775, 11124674379405792463701519059

Offset: 0

Peter Bala, Mar 09 2015

A000364(n) = (-1)^n*2^(2*n)*Euler(2*n,1/2), where E(n,x) is the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255882 (k = 3), A255883(k = 4) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A000364, A188514, A255882, A255883, A255884.

#A255881
k := 2:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 16)): seq(coeftayl(%, x = 0, n), n = 0 .. 16);

A000364:= Table[Abs[EulerE[2 n]], {n, 0, 80}]; a:= With[{nmax = 70}, CoefficientList[Series[Exp[Sum[A000364[[k + 1]]*x^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( x + 5*x^2/2 + 61*x^3/3 + 1385*x^4/4 + ... ) = 1 + x + 3*x^2 + 23*x^3 + 371*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*2^(2*k)*E(2*k,1/2)*a(n-k).
a(n) ~ 2^(4*n + 3) * n^(2*n - 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019

A188458 Expansion of e.g.f. exp(x)/cosh(2*x).

Original entry on oeis.org

1, 1, -3, -11, 57, 361, -2763, -24611, 250737, 2873041, -36581523, -512343611, 7828053417, 129570724921, -2309644635483, -44110959165011, 898621108880097, 19450718635716001, -445777636063460643, -10784052561125704811, 274613643571568682777

Offset: 0

Paul D. Hanna, Apr 01 2011

sign

A signed version of A001586 (Springer numbers).

Equals the logarithmic derivative of A188514 (ignoring the initial term of this sequence); note that the unsigned version (A001586) does not form a logarithmic derivative of an integer sequence.

			E.g.f.: exp(x)/cosh(2*x) = 1 + x - 3*x^2/2! - 11*x^3/3! + 57*x^4/4! + 361*x^5/5! +...
Illustration of other generating functions.
E.g.f.: 1 = exp(-x) + exp(x)*x - 3*exp(-x)*x^2/2! - 11*exp(x)*x^3/3! +...
L.g.f.: log(1+x) = x/(1-x) - 3*(x^2/2)/(1+x)^2 - 11*(x^3/3)/(1-x)^3 +...
G.f.: 1 = 1/(1+x) + 1*x/(1-x)^2 - 3*x^2/(1+x)^3 - 11*x^3/(1-x)^4 +...
G.f.: 1 = 1/(1+x)^2 + 1*2*x/(1-x)^3 - 3*3*x^2/(1+x)^4 - 11*4*x^3/(1-x)^5 +...
G.f.: 1 = 1/(1+x)^3 + 1*3*x/(1-x)^4 - 3*6*x^2/(1+x)^5 - 11*10*x^3/(1-x)^6 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001586, A188514.

seq(4^n*euler(n,3/4), n=0..20); # Peter Luschny, Apr 19 2014

CoefficientList[Series[E^x/Cosh[2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 07 2013 *)

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp(X)/cosh(2*X),n)}

{a(n)=n!*polcoeff(1-sum(k=0, n-1, a(k)*exp(-(-1)^k*x+x*O(x^n))*x^k/k!), n)}

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+(-1)^k*x+x*O(x^n))^(k+1)), n)}

/* Holds for m>=1: */
{a(n)=local(m=1); polcoeff(1-sum(k=0, n-1, a(k)*binomial(m+k-1, k)*x^k/(1+(-1)^k*x+x*O(x^n))^(k+m)), n)/binomial(m+n-1, n)}

/* Recurrence: */
{a(n)=if(n<0,0,if(n==0,1, sum(k=1, n, -(-1)^(n*k)*binomial(n, k)*a(n-k))))}

{EULER(n)=n!*polcoeff(1/cosh(x+x*O(x^n)),n)}
{a(n)=sum(k=0,n,2^k*binomial(n,k)*EULER(k))}

{a(n)=(-1)^(n\2)*((1+I)/2)^n*sum(k=0, n, ((1-I)/(1+I))^k*sum(j=0, k, (-1)^(k-j)*binomial(n+1, k-j)*(2*j+1)^n))}

a(n) = Sum_{k=1..n} -(-1)^(n*k)*C(n, k)*a(n-k) for n>0 with a(0)=1.
L.g.f.: log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + (-1)^n*x)^n.
E.g.f.: 1 = Sum_{n>=0} a(n)*exp(-(-1)^n*x)*x^n/n!.
G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + (-1)^n*x)^(n+1).
G.f.: 1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + (-1)^n*x)^(n+m) for m>=1.
a(n) = Sum_{k=0..n} 2^k C(n,k) Euler(k). - Peter Luschny
a(n) = (-1)^[n/2]*((1+I)/2)^n * Sum_{k=0..n} ((1-I)/(1+I))^k * Sum_{j=0..k} (-1)^(k-j)*C(n+1, k-j)*(2*j+1)^n. - Peter Bala
O.g.f.: 1/(1-x/(1+4*x/(1-x- 4*x/(1+4*x/(1+x- 6*x/(1+6*x/(1+x- 8*x/(1+8*x/(1+x- 10*x/(1+10*x/(1+x- 12*x/(1+12*x/(1+x- ...))))))))))))) (continued fraction).
E.g.f.: E(x) = exp(x)/cosh(2*x) = 2/G(0) where G(k)= 1 -((-1)^k)*3^k/(1 - x/(x + (k+1)*((-1)^k)*3^k/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jun 07 2012
a(n) ~ n! * (cos(n*Pi/2) + sin(n*Pi/2)) * 2^(2*n+3/2) / Pi^(n+1). - Vaclav Kotesovec, Oct 07 2013
G.f.: conjecture T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 + (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
From Peter Luschny, Apr 19 2014: (Start)
a(n) = 2^n*skp(n, 1/2), where skp(n,x) are the Swiss-Knife polynomials A153641.
a(n) = 4^n*E(n, 3/4), where E(n,x) are Euler polynomials.
a(n) = (8^n/((n+1)/2))*(B(n+1, 7/8) - B(n+1, 3/8)), where B(n,x) are the Bernoulli polynomials. (End)
a(n) = 2^(3*n+1)*(Zeta(-n,3/8)-Zeta(-n,7/8)). - Peter Luschny, Oct 15 2015

A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).

Original entry on oeis.org

1, -1, -3, 11, 57, -361, -2763, 24611, 250737, -2873041, -36581523, 512343611, 7828053417, -129570724921, -2309644635483, 44110959165011, 898621108880097, -19450718635716001, -445777636063460643, 10784052561125704811, 274613643571568682777

Offset: 0

Michael Somos, Jun 21 2012

sign

			G.f. = 1 - x - 3*x^2 + 11*x^3 + 57*x^4 - 361*x^5 - 2763*x^6 + 24611*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001586, A046717, A188458, A188514, A255883.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x)/Cosh(2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 10 2018

CoefficientList[Series[2*E^x/(E^(4*x)+1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ -x] / Cosh[ 2 x], {x, 0, n}]]; (* Michael Somos, Aug 26 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( exp(-x + A) / cosh( 2*x + A), n))};

@CachedFunction
def p(n,x) :
    if n == 0 : return 1
    w = -1 if n%2 == 0 else  0
    v =  1 if n%2 == 0 else -1
    return v*add(p(k,0)*binomial(n,k)*(x^(n-k)+w) for k in range(n)[::2])
def A212435(n) : return 2^n*p(n, 1/2)
[A212435(n) for n in (0..20)]  # Peter Luschny, Jul 19 2012

E.g.f.: 2 * exp(x) / (exp(4*x) + 1).
E.g.f. is the reciprocal of the e.g.f. of A046717.
a(n) = (-1)^n * A188458(n) = (-1)^floor((n + 1) / 2) * A001586(n).
E.g.f.: 2/E(0), where E(k) = 1 + (-1)^k/(3^k - 3*9^k*x/(3*3^k*x + (-1)^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
G.f.: conjecture T(0)/(1+x), where T(k) = 1 - 4*x^2*(k+1)^2/(4*x^2*(k+1)^2 + (1+ x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013
a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) * 2^(2*n+3/2) / Pi^(n+1). - Vaclav Kotesovec, Feb 25 2014
From Peter Bala, Mar 10 2015: (Start)
a(n) = 4^n*E(n,1/4).
O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - x*(4*k + 1)).
The series expansion exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 - x - x^2 + 5*x^3 + 11*x^4 - 91*x^5 - 391*x^6 + ... appears to have integer coefficients. Cf. A188514, A255883. (End)

A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

Original entry on oeis.org

1, 2, 13, 224, 8170, 522716, 51749722, 7309866728, 1394040714169, 344865267322010, 107361980072755261, 41067497940750566312, 18931745446455458282248, 10350955324610065848650384, 6622526747212249020075069880, 4901565185965701578921602882976

Offset: 0

Peter Bala, Mar 09 2015

A210657(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial. In general it appears that when is k a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255883(k = 4) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A188514, A210657, A255881, A255883, A255884.

#A255882
k := 3:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A210657[n_]:= 9^n EulerE[2 n, 1/3]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A210657[k]*(-x)^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 51}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 2*x + 22*x^2/2 + 602*x^3/3 + 30742*x^4/4 + ... ) = 1 + 2*x + 13*x^2 + 224*x^3 + 8170*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*3^(2*k)*E(2*k,1/3)*a(n-k).
a(n) ~ 2^(2*n + 2) * 3^(2*n + 1/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

A255883 Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).

Original entry on oeis.org

1, 3, 33, 1011, 65985, 7536099, 1329205857, 334169853267, 113370124235649, 49880529542872515, 27614111852126579361, 18782012442066306225843, 15394836674855296870428993, 14965462261283347594195897251, 17023467576167762236198869304545, 22400927665017118737825435362462739

Offset: 0

Peter Bala, Mar 09 2015

A000281(n) =(-1)^n*4^(2*n)*E(2*n,1/4), where E(n,x) denotes the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255882(k = 3) and A255884 (k = 6).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A000281, A188514, A255881, A255882, A255884

#A255883
k := 4:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A000281:= With[{nn = 200}, Take[CoefficientList[Series[Cos[x]/Cos[2 x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A000281[[k + 1]]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}]  (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 3*x + 57*x^2/2 + 2763*x^3/3 + 250737*x^4/4 + ... ) = 1 + 3*x + 33*x^2 + 1011*x^3 + 65985*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*4^(2*k)*E(2*k,1/4)*a(n-k).
a(n) ~ 2^(6*n + 5/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

A255884 Expansion of exp( Sum_{n >= 1} A002438(n)*x^n/n ).

Original entry on oeis.org

1, 5, 115, 7955, 1179715, 304888655, 121350927565, 68751844662605, 52528700295424915, 52031089992310711055, 64835758857480094584265, 99249388572274155967996505, 183075972804988649078529524365, 400493686169423616676960341062705, 1025151296160300228944197705742007715

Offset: 0

Peter Bala, Mar 09 2015

A002438(n+1) =(-1)^n*6^(2*n)*E(2*n,1/6), where E(n,x) denotes the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255882(k = 3) and A255883 (k = 4).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A002438, A188514, A255881, A255882, A255883.

#A255884
k := 6:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 14)): seq(coeftayl(%, x = 0, n), n = 0 .. 14);

A000243[n_]:= (1 + 9^(n - 1))*Abs[EulerE[2*(n - 1)]]/2; a:= With[{nmax = 75}, CoefficientList[Series[Exp[Sum[A000243[k + 1]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 5*x + 205*x^2/2 + 22265*x^3/3 + 4544185 *x^4/4 + ... ) = 1 + 5*x + 115*x^2 + 7955*x^3 + 1179715*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*6^(2*k)*E(2*k,1/6)*a(n-k).
a(n) ~ 2^(4*n + 2) * 3^(2*n) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

A092635 Coefficients in asymptotic expansion of (Pin/16^n)binomial(2n,n)^2 in powers of 1/(8n).

Original entry on oeis.org

1, -2, 2, 4, -10, -92, 212, 4744, -10330, -440492, 926972, 64905784, -133989700, -13958115992, 28507072232, 4129849252624, -8382195909370, -1609609189734092, 3254514296768492, 799400112196210264, -1612314122867559340

Offset: 0

Michael Somos, Mar 02 2004

sign

			binomial(2*n,n)^2*Pi*n/16^n ~ 1 -2/(8n) +2/(8n)^2 +4/(8n)^3 + ...
From _Paul D. Hanna_, Apr 02 2011: (Start)
O.g.f.: A(x) = 1 - 2*x + 2*x^2 + 4*x^3 - 10*x^4 - 92*x^5 + 212*x^6 + ...
E.g.f.: E(x) = 1 - 2*x + 2*x^2/2! + 4*x^3/3! - 10*x^4/4! - 92*x^5/5! + 212*x^6/6! + ...
Note that E(x)*exp(2*x) = sqrt(E(x)*E(-x)) is an even function:
E(x)*exp(2*x) = 1 - 2*x^2/2! + 22*x^4/4! - 692*x^6/6! + 45862*x^8/8! + ... (End)

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A002894, A012393, A188514.

#Uses the conjectural recurrence equation
A092635 := proc (n) option remember; if n = 0 then 1 else add((-4)^(n-k)*euler(n-k, 1)*A092635(k), k = 0 .. n-1)/n end if; end proc:
seq(A092635(n), n = 0 .. 20); # Peter Bala, Mar 12 2015

m = 21;
1/(1+2x/(1-x+ContinuedFractionK[(4i^2-1)x^2, 1, {i, 1, m/2 // Floor}]))+O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)
CoefficientList[Normal[Series[Pi*n*Binomial[n/4, n/8]^2 / 2^(n/2+3), {n, Infinity, 20}]], 1/n] (* Vaclav Kotesovec, Aug 18 2024 *)

a(n)=if(n<0,0,polcoeff(exp(intformal(serlaplace(-1/cosh(x*2+x*O(x^n))^2*2))),n))

a(n)=local(A);if(n<1,n==0,A=1; for(m=1,n,A+=x^2*O(x^m);A=Pol(A)+polcoeff(subst(A,x,x/(1+8*x))/A*(1+8*x)/(1+4*x)^2,m+1)/8/m*x^m);polcoeff(A,n))

{B(x,n) = prod(k=1,n, (1 + 4*k*x)^((-1)^k*binomial(n,k)) )}
{a(n) = polcoeff( A = prod(m=1,n, B(x  +x*O(x^n), m)^(1/2^m)), n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Mar 03 2024

E.g.f. E(x) satisfies: E(x) = E(-x)*exp(-4*x). - Paul D. Hanna, Apr 02 2011
O.g.f.: A(x) = 1/(1 + 2*x/(1-x + 3*x^2/(1 + 15*x^2/(1 + 35*x^2/(1 + 63*x^2/(1 + 99*x^2/(1 + ...+ (4*n^2-1)*x^2/(1 + ...)))))))) (continued fraction). - Paul D. Hanna, Apr 02 2011
From Peter Bala, Mar 12 2015: (Start)
Let 2/(1 + exp(4*x)) = Sum_{n >= 0} b(n)*x^n/n!, so that b(n) = (-4)^n*E(n,1), where E(n,x) is the n-th Euler polynomial.
Then exp( Sum_{n >= 1} b(n)*x^n/n ) = 1 - 2*x + 2*x^2 + 4*x^3 - 10*x^4 - 92*x^5 + ... appears to be the o.g.f. for this sequence.
Assuming this is true we have the recurrence a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} (-4)^(n-k)*E(n-k,1)*a(k) for n >= 1. (End)
O.g.f.: A(x) = Product_{n>=1} B(x,n)^(1/2^n) where B(x,n) = Product{k=1..n} (1 + 4*k*x)^((-1)^k*binomial(n,k)). Compare to Product_{n>=1} B(x,n) = 1 - 4*x and Product_{n>=1} B(x,n)^(1/n) = exp(-4*x). - Paul D. Hanna, Mar 03 2024
Conjecture from Paul D. Hanna, Aug 15 2024: (Start)
Let A(x) = Sum_{n>=0} a(n)*x^(n+1), then A(x) satisfies:
  A(x) = -x^2 / A(-x),
  A(x) = -A(-x/(1+4*x)),
  A(x) = x^2 / A(x/(1-4*x)). (End)

A255900 Expansion of exp( Sum_{n >= 1} A000464(n-1)*x^n/n ).

Original entry on oeis.org

1, 1, 6, 126, 6291, 581499, 86010084, 18599726484, 5532984567639, 2166830287921959, 1080602568966548022, 668603866168566179982, 502601850887023486736757, 451167540495577093495790397, 476690705981608679350490956032, 585587125411920596898761442409728

Offset: 0

Peter Bala, Mar 10 2015

For similar results see A188514 and A255881.
A000464(n-1) = (-1)^n*4^(2*n-1)*E(2*n-1,1/4), where E(n,x) denotes the n-th Euler polynomial.
More generally, calculation suggests that for integer h and a nonzero integer k the expansion of exp ( Sum_{n >= 1} (2*k)^(2*n-1)*E(2*n-1,h/(2*k)) )*x^n/n has integer coefficients. This is the case h = 1 and k = 2.

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000464, A188514, A255881.

#A255900
k := 2:
exp(add((2*k)^(2*n-1)*euler(2*n-1, 1/(2*k))*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

A000464:= With[{nn = 200}, Take[CoefficientList[Series[Sin[x]/Cos[2 x], {x, 0, nn}], x]*Range[0, nn-1]!, {2, -1, 2}]]; a:= With[{nmax = 50}, CoefficientList[Series[Exp[x + Sum[A000464[[k]]*x^(k)/(k), {k, 2, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( x + 11*x^2/2 + 361*x^3/3 + 24611*x^4/4 + ... ) = 1 + x + 6*x^2 + 126*x^3 + 6291*x^4 + ....

a(0) = 1 and a(n) = 1/n*Sum_{k = 1..n} (-1)^k*4^(2*k-1)*E(2*k-1,1/4)*a(n-k) for n >= 1.

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A255881 Expansion of exp( Sum_{n >= 1} A000364(n)*x^n/n ).

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A188458 Expansion of e.g.f. exp(x)/cosh(2*x).

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A212435 Expansion of e.g.f.: exp(-x) / cosh(2*x).

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A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

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A255883 Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).

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A255884 Expansion of exp( Sum_{n >= 1} A002438(n)*x^n/n ).

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A092635 Coefficients in asymptotic expansion of (Pi*n/16^n)*binomial(2n,n)^2 in powers of 1/(8n).

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A092635 Coefficients in asymptotic expansion of (Pin/16^n)binomial(2n,n)^2 in powers of 1/(8n).