A218300 -id:A218300 - cp's OEIS frontend

A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.

1, 1, 4, 38, 576, 12052, 322848, 10564304, 408903680, 18288706544, 928575662400, 52780935007968, 3321208845997056, 229232635832433664, 17221699990084108288, 1399139700462119135232, 122235936429355565580288, 11428226675376971405577984, 1138551595285580854471388160

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} (n+1)^(n-1) * exp(-(n+1)*x) * x^n/n!.
More generally, if we define a(n) for fixed integers m, t, and s>=0, by:
(0) Sum_{n>=0} m * n^(s*n) * (n*t+m)^(n-1) * exp(-n^s*(n*t+m)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n
then the coefficients a(n) are integral and may be expressed by:
(1) a(n) = 1/n! * Sum_{k=0..n} m*(-1)^(n-k)*binomial(n,k) * k^(s*n) * (k*t+m)^(n-1).
(2) a(n) = 1/n! * [x^n] Sum_{k>=0} m*k^(s*k)*(k*t+m)^(k-1)*x^k / (1 + k^s*(k*t+m)*x)^(k+1).
(3) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1+m*x)^(n-1) / Product_{k=1..n} (1-k*t*x).
(4) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1-m*x)^(s*n) / Product_{k=1..n} (1-(k*t+m)*x).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 576*x^4 + 12052*x^5 + 322848*x^6 +...
where
A(x) = 1 + 1^1*2^0*x*exp(-1*2*x) + 2^2*3^1*exp(-2*3*x)*x^2/2! + 3^3*4^2*exp(-3*4*x)*x^3/3! + 4^4*5^3*exp(-4*5*x)*x^4/4! + 5^5*6^4*exp(-5*6*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A078739, A217899, A217901, A217902, A217903, A217904, A217905, A217910, A217913, A218300.

a[n_] := 1/n!*Sum[(-1)^(n-k)*Binomial[n, k]*k^n*(k+1)^(n-1), {k, 0, n}]; a[0]=1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 06 2013 *)

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

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

{a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^n*(k+1)^(n-1))}

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

{a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(k+1)*x +x*O(x^n)), n)}
for(n=0,30,print1(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=0,n-1, binomial(n-1,k) * Stirling2(2*n-k-1,n)))} \\ Paul D. Hanna, Nov 13 2012
/* PARI Programs for the General Case (START) ...................... */

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

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

{a(n,m=1,t=1,s=1)=1/n!*sum(k=0, n, m*(-1)^(n-k)*binomial(n, k)*k^(s*n)*(t*k+m)^(n-1))}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1+m*x)^(n-1)/prod(k=0, n, 1-t*k*x +x*O(x^(s*n))), s*n)}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1-m*x)^(s*n)/prod(k=0, n, 1-(t*k+m)*x +x*O(x^(s*n))), s*n)}
/* (END) ........................................................... */

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+1)^(k-1)*x^k / (1 + k*(k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1-(k+1)*x).
a(n) = A078739(n,n) for n>=1.
a(n) = Sum_{k=0..n-1} binomial(n-1,k) * Stirling2(2*n-k-1,n) for n>0, where Stirling2(n,k) = A008277(n,k). - Paul D. Hanna, Nov 13 2012
a(n) ~ 2^(2*n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, May 09 2014

A201595 E.g.f. satisfies A(x) = exp(xA(x)) cosh(x*A(x)).

Original entry on oeis.org

1, 1, 4, 28, 288, 3936, 67328, 1385728, 33372160, 921118720, 28677169152, 994360565760, 38007586684928, 1587878686621696, 71990467473965056, 3520403893852831744, 184707311409882464256, 10350444842488122310656, 616975843658373414256640, 38981881007475178476666880

Offset: 0

Paul D. Hanna, Dec 03 2011

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! +...
The coefficients of x^n/n! in initial powers of G(x) = (1 + exp(2*x))/2 begin:
G^1: [(1), 1, 2, 4, 8, 16, 32, 64, 128, ...];
G^2: [1,(2), 6, 20, 72, 272, 1056, 4160, ...];
G^3: [1, 3,(12), 54, 264, 1368, 7392, 41184, ...];
G^4: [1, 4, 20,(112), 680, 4384, 29600, 207232, ...];
G^5: [1, 5, 30, 200,(1440), 11000, 88080, 732800, ...];
G^6: [1, 6, 42, 324, 2688,(23616), 217392, 2080224, ...];
G^7: [1, 7, 56, 490, 4592, 45472,(471296), 5076400, ...];
G^8: [1, 8, 72, 704, 7344, 80768, 928512,(11085824), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 112/4, 1440/5, 23616/6, 471296/7, 11085824/8, ...].

G. C. Greubel, Table of n, a(n) for n = 0..370
Nantel Bergeron, Laura Colmenarejo, Shu Xiao Li, John Machacek, Robin Sulzgruber, Mike Zabrocki, Adriano Garsia, Marino Romero, Don Qui, and Nolan Wallach, Super Harmonics and a representation theoretic model for the Delta conjecture, A summary of the open problem sessions of Jan 24, 2019, Representation Theory Connections to (q,t)-Combinatorics (19w5131), Banff, BC, Canada.
John Lentfer, A conjectural basis for the (1,2)-bosonic-fermionic coinvariant ring, arXiv:2406.19715 [math.CO], 2024. See p. 2.

Cf. A214225, A201594, A074932, A218300, A218301, A218302.

Cf. A370907, A370908.

Join[{1},Table[Sum[Binomial[n+1,k] k^n/(n+1),{k,0,n+1}]/2,{n,20}]] (* Harvey P. Dale, Feb 04 2012 *)
CoefficientList[Series[(x-LambertW[-x*E^x])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 04 2012 *)

a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)

a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)

a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)

/* Formula for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!: */
{a(n,m=1)=sum(k=0, n+m, binomial(n+m, k)*k^n*m/(n+m)/2^m)}

/* Formula derived from a LambertW identity: */
{a(n)=local(A=sum(k=0,n,(k+1)^(k-1)*cosh((k+1)*x+x*O(x^n))*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Oct 24 2012

/* Formula derived from a LambertW identity: */
{a(n)=local(A=1+sum(k=1,n,k^k*sinh(k*x+x^2*O(x^n))/(k*x)*x^k/k!));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

a(n) = (1/2) * Sum_{k=0..n+1} C(n+1,k) * k^n / (n+1).
a(n) = [x^n/n!] exp((n+1)*x) * cosh(x)^(n+1) / (n+1).
E.g.f. A(x) satisfies:
(1) A( x*exp(-x)/cosh(x) ) = exp(x)*cosh(x).
(2) A(x) = (1/x)*Series_Reversion( x*exp(-x)/cosh(x) ).
(3) A(x) = (1 + exp(2*x*A(x)))/2.
(4) A(x) = exp(G(x)) where G(x) is the e.g.f. of A074932.
(5) A(x) = Sum_{n>=0} (n+1)^(n-1) * cosh((n+1)*x) * x^n/n!. - Paul D. Hanna, Oct 24 2012
(6) A(x) = 1 + Sum_{n>=1} n^n * sinh(n*x)/(n*x) * x^n/n!. - Paul D. Hanna, Nov 20 2012
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = Sum_{k=0..n+m} C(n+m, k) * k^n * m/(n+m) / 2^m.
a(n) = A214225(n+1)/(n+1).
E.g.f.: (x-LambertW(-x*exp(x)))/(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!*sqrt(LambertW(exp(-1))+1)/(2*sqrt(2*Pi)*n^(3/2)*LambertW(exp(-1))^(n+1)). - Vaclav Kotesovec, Dec 04 2012
G.f.: 1/2 + 1/2 * Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1). - Seiichi Manyama, Apr 23 2024
a(n) = n! * Sum_{k=0..n} 2^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

Original entry on oeis.org

2, 7, 8, 4, 6, 4, 5, 4, 2, 7, 6, 1, 0, 7, 3, 7, 9, 5, 1, 0, 9, 3, 5, 8, 7, 3, 9, 0, 2, 2, 9, 8, 0, 1, 5, 5, 4, 3, 9, 4, 7, 7, 4, 8, 8, 6, 1, 9, 7, 4, 5, 7, 6, 5, 4, 5, 3, 1, 7, 8, 1, 0, 5, 5, 3, 5, 0, 2, 9, 3, 7, 5, 4, 5, 9, 9, 4, 9, 8, 9, 8, 1, 9, 2, 0, 4, 9, 8, 4, 2, 8, 1, 1, 2, 9, 9, 4, 2, 8

Offset: 0

Clark Kimberling, Dec 18 2011

See A202322 for a guide to related sequences. The Mathematica program includes a graph.

			0.2784645427610737951093587390229801554394774886...

Heine Halberstam and Hans Egon-Richert, Sieve Methods, Dover Publications (2011). See Theorem 2.1.

G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Gautschi, The incomplete Gamma Function since Tricomi, Atti Conv. Lincei 147 (1999) 203, eq. (2.16).
H. J. H. Tuenter, On the generalized Poisson distribution, arXiv:math/0606238 [math.ST], 2006. Published version on On the Generalized Poisson Distribution, Statistica Neerlandica, 54(3):374-376, November 2000.
S. M. Zemyan, On the zeros of the Nth partial sum of the exponential series, Am. Math. Monthly 112 (2005) 891-909.
Index entries for transcendental numbers.

Cf. A202322, A215077, A218300.

u = E; v = 0;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .27, .28}, WorkingPrecision -> 110]
RealDigits[r]  (* A202357 *)
RealDigits[ ProductLog[1/E], 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)
RealDigits[LambertW[Exp[-1]],10,120][[1]] (* Harvey P. Dale, Dec 24 2019 *)

lambertw(exp(-1)) \\ Michel Marcus, Mar 21 2016

The constant in A202355 minus 1. - R. J. Mathar, Dec 21 2011
1+x+log(x)=0. - R. J. Mathar, Nov 02 2012
Equals LambertW(exp(-1)). - Vaclav Kotesovec, Jan 10 2014

A218301 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(3x)cosh(3x).

Original entry on oeis.org

1, 3, 24, 252, 3360, 55008, 1074816, 24499968, 639744000, 18856765440, 619897847808, 22502300590080, 894419152404480, 38651030120693760, 1804765006764441600, 90574514900736933888, 4862862027933962207232, 278158492957848901779456, 16889663645642083220324352

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + 3*x + 24*x^2/2! + 252*x^3/3! + 3360*x^4/4! + 55008*x^5/5! +...
where
A(x) = cosh(3*x) + 3*4^0*cosh(4*x)*x + 3*5^1*cosh(5*x)*x^2/2! + 3*6^2*cosh(6*x)*x^3/3! + 3*7^3*cosh(7*x)*x^4/4! + 3*8^4*cosh(8*x)*x^5/5! +...

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A201595, A218300, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(k+3)^(k-1)*cosh((k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} 3*(n+3)^(n-1) * cosh((n+3)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3*LambertW(-x * exp(x)) ).
a(n) = 3/2 * Sum_{k=0..n} (k+3)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 3/2 * Sum_{k>=0} (k+3)^(k-1) * x^k/(1 - (k+3)*x)^(k+1). (End)

A218302 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(4x)cosh(4x).

Original entry on oeis.org

1, 4, 40, 496, 7488, 134784, 2836736, 68635648, 1881948160, 57777184768, 1965962575872, 73503311167488, 2997314388623360, 132455836580577280, 6308164435588415488, 322185156718017642496, 17571327124936467677184, 1019377026461494381903872

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + 4*x + 40*x^2/2! + 496*x^3/3! + 7488*x^4/4! +...
where
A(x) = cosh(4*x) + 4*5^0*cosh(5*x)*x + 4*6^1*cosh(6*x)*x^2/2! + 4*7^2*cosh(7*x)*x^3/3! + 4*8^3*cosh(8*x)*x^4/4! + 4*9^4*cosh(9*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(4*R)*cosh(4*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,4*(k+4)^(k-1)*cosh((k+4)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} 4*(n+4)^(n-1) * cosh((n+4)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 4*x - 4*LambertW(-x * exp(x)) ).
a(n) = 2 * Sum_{k=0..n} (k+4)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 2 * Sum_{k>=0} (k+4)^(k-1) * x^k/(1 - (k+4)*x)^(k+1). (End)

A218303 E.g.f. A(x) satisfies A( x/(exp(2x)cosh(2x)) ) = exp(x)cosh(x).

Original entry on oeis.org

1, 1, 6, 76, 1480, 39056, 1303904, 52716224, 2504292480, 136741146880, 8439125550592, 580959483530240, 44138582550333440, 3668643339883089920, 331143571990522060800, 32258185015683531587584, 3373221864252806213435392, 376881845889001869159759872

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 76*x^3/3! + 1480*x^4/4! + 39056*x^5/5! +...
where
A(x) = cosh(x) + 3^0*cosh(3*x)*x + 5^1*cosh(5*x)*x^2/2! + 7^2*cosh(7*x)*x^3/3! + 9^3*cosh(9*x)*x^4/4! + 11^4*cosh(11*x)*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A201595, A218300, A218301, A218302, A218304, A218305, A218306, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(2*X)*cosh(2*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(2*k+1)^(k-1)*cosh((2*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} (2*n+1)^(n-1) * cosh((2*n+1)*x) * x^n/n!.
a(n) ~ c * 2^n * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/(4*sqrt(LambertW(exp(-1)))) = 0.535672560704567808218663129282561449... . - Vaclav Kotesovec, Jul 13 2014, updated Jun 10 2019
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/2 * LambertW(-2*x * exp(2*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (2*k+1)^(k-1) * x^k/(1 - (2*k+1)*x)^(k+1). (End)

A218304 E.g.f. A(x) satisfies A( x/(exp(2x)cosh(2x)) ) = exp(3x)cosh(3x).

Original entry on oeis.org

1, 3, 30, 468, 10248, 291888, 10282464, 432631104, 21195292800, 1186054914816, 74676568432128, 5226914768016384, 402722750814750720, 33876716756962652160, 3089713688099323502592, 303723970839738425622528, 32015024916407062538256384

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + 3*x + 30*x^2/2! + 468*x^3/3! + 10248*x^4/4! + 291888*x^5/5! +...
where
A(x) = cosh(3*x) + 3*5^0*cosh(5*x)*x + 3*7^1*cosh(7*x)*x^2/2! + 3*9^2*cosh(9*x)*x^3/3! + 3*11^3*cosh(11*x)*x^4/4! + 3*13^4*cosh(13*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218302, A218303, A218305, A218306, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(2*X)*cosh(2*X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(2*k+3)^(k-1)*cosh((2*k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} 3*(2*n+3)^(n-1) * cosh((2*n+3)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/2 * LambertW(-2*x * exp(2*x)) ).
a(n) = 3/2 * Sum_{k=0..n} (2*k+3)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 3/2 * Sum_{k>=0} (2*k+3)^(k-1) * x^k/(1 - (2*k+3)*x)^(k+1). (End)

A218305 E.g.f. A(x) satisfies A( x/(exp(3x)cosh(3x)) ) = exp(x)cosh(x).

Original entry on oeis.org

1, 1, 8, 148, 4256, 166816, 8297600, 500730112, 35547379712, 2902899914752, 268094176428032, 27629598827044864, 3143573312615481344, 391375817676973932544, 52926434374336385122304, 7725597721066205089890304, 1210677595048894480252928000

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 148*x^3/3! + 4256*x^4/4! + 166816*x^5/5! +...
where
A(x) = cosh(x) + 4^0*cosh(4*x)*x + 7^1*cosh(7*x)*x^2/2! + 10^2*cosh(10*x)*x^3/3! + 13^3*cosh(13*x)*x^4/4! + 16^4*cosh(16*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218306, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(3*k+1)^(k-1)*cosh((3*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} (3*n+1)^(n-1) * cosh((3*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/3 * LambertW(-3*x * exp(3*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(1 - (3*k+1)*x)^(k+1). (End)

A218306 E.g.f. A(x) satisfies A( x/(exp(3x)cosh(3x)) ) = exp(2x)cosh(2x).

Original entry on oeis.org

1, 2, 20, 392, 11648, 466112, 23517824, 1434077696, 102618951680, 8432793964544, 782753794531328, 81007725700038656, 9249066952457584640, 1154952975718091325440, 156588371428134115868672, 22908199202756436344963072, 3597006040171205977538822144

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x), then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 392*x^3/3! + 11648*x^4/4! + 466112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*5^0*cosh(5*x)*x + 2*8^1*cosh(8*x)*x^2/2! + 2*11^2*cosh(11*x)*x^3/3! + 2*14^3*cosh(14*x)*x^4/4! + 2*17^4*cosh(17*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218307, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,2*(3*k+2)^(k-1)*cosh((3*k+2)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} 2*(3*n+2)^(n-1) * cosh((3*n+2)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2/3 * LambertW(-3*x * exp(3*x)) ).
a(n) = Sum_{k=0..n} (3*k+2)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + Sum_{k>=0} (3*k+2)^(k-1) * x^k/(1 - (3*k+2)*x)^(k+1). (End)

A218307 E.g.f. A(x) satisfies A( x/(exp(4x)cosh(4x)) ) = exp(x)cosh(x).

Original entry on oeis.org

1, 1, 10, 244, 9288, 483216, 31949216, 2564959552, 242374510720, 26355555496192, 3241906046249472, 445085008158569472, 67469834196870809600, 11192986206960277688320, 2017105871358529382883328, 392394481517424330142203904, 81955683182673295403291541504

Offset: 0

Paul D. Hanna, Oct 25 2012

nonn

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 244*x^3/3! + 9288*x^4/4! + 483216*x^5/5! +...
where
A(x) = cosh(x) + 5^0*cosh(5*x)*x + 9^1*cosh(9*x)*x^2/2! + 13^2*cosh(13*x)*x^3/3! + 17^3*cosh(17*x)*x^4/4! + 21^4*cosh(21*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218308, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

/* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(4*k+1)^(k-1)*cosh((4*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: A(x) = Sum_{n>=0} (4*n+1)^(n-1) * cosh((4*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/4 * LambertW(-4*x * exp(4*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (4*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(1 - (4*k+1)*x)^(k+1). (End)

cp's OEIS Frontend

A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n*(n+1)*x) * x^n / n!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

PARI

PARI

PARI

PARI

PARI

Formula

A201595 E.g.f. satisfies A(x) = exp(x*A(x)) * cosh(x*A(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

Formula

A202357 Decimal expansion of the number x satisfying e*x = e^(-x).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A218301 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(3*x)*cosh(3*x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A218302 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(4*x)*cosh(4*x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A218303 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(x)*cosh(x).

Views

Author

Keywords

Comments

A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.

A201595 E.g.f. satisfies A(x) = exp(xA(x)) cosh(x*A(x)).

A218301 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(3x)cosh(3x).

A218302 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(4x)cosh(4x).

A218303 E.g.f. A(x) satisfies A( x/(exp(2x)cosh(2x)) ) = exp(x)cosh(x).

A218304 E.g.f. A(x) satisfies A( x/(exp(2x)cosh(2x)) ) = exp(3x)cosh(3x).

A218305 E.g.f. A(x) satisfies A( x/(exp(3x)cosh(3x)) ) = exp(x)cosh(x).

A218306 E.g.f. A(x) satisfies A( x/(exp(3x)cosh(3x)) ) = exp(2x)cosh(2x).

A218307 E.g.f. A(x) satisfies A( x/(exp(4x)cosh(4x)) ) = exp(x)cosh(x).