A201595 -id:A201595 - cp's OEIS frontend

A218300 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(2x)cosh(2x).

1, 2, 12, 104, 1216, 18112, 329600, 7108096, 177549312, 5046554624, 160947232768, 5694342479872, 221410157133824, 9387011838312448, 431051678297358336, 21316106766591721472, 1129526392342026649600, 63855305138514241257472, 3836490516381680506241024

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 + 12*x^2/2! + 104*x^3/3! + 1216*x^4/4! + 18112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*3^0*cosh(3*x)*x + 2*4^1*cosh(4*x)*x^2/2! + 2*5^2*cosh(5*x)*x^3/3! + 2*6^3*cosh(6*x)*x^4/4! + 2*7^4*cosh(7*x)*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A201595, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310, A217900.

Cf. A202357.

nmin = 0; nmax = 18; sol = {a[0] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[a[k] x^k/k!, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/(Exp[x] Cosh[x])] - Exp[2x] Cosh[2x] + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol+1, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(X)*cosh(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*(k+2)^(k-1)*cosh((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*(n+2)^(n-1) * cosh((n+2)*x) * x^n/n!.
E.g.f.: A(x) = 1 + Sum_{n>=0} 2*(n+2)^(n-1) * sinh((n+2)*x) * x^n/n!.
a(n) ~ c * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/LambertW(exp(-1))^2 = 14.5815783688217906961670551786416446... . - 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( 2*x - 2*LambertW(-x * exp(x)) ).
a(n) = Sum_{k=0..n} (k+2)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + Sum_{k>=0} (k+2)^(k-1) * x^k/(1 - (k+2)*x)^(k+1). (End)

A349562 Number of labeled rooted forests with 2-colored leaves.

Original entry on oeis.org

1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760

Offset: 0

Alexander Burstein, Nov 22 2021

nonn

a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).

			a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).

Seiichi Manyama, Table of n, a(n) for n = 0..372
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000007, A007889, A216857, A038049, A201595,

CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 25 2021 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Nov 26 2021

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
a(n) = A216857(n+1)/(n+1).
a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.
a(n) = 2*A201595(n) - A000007(n).
E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).
E.g.f. satisfies: A(-x*A(x)) = 1/A(x).
From Vaclav Kotesovec, Nov 25 2021: (Start)
E.g.f.: -LambertW(-x*exp(x))/x.
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).
(End)
From Seiichi Manyama, Nov 26 2021: (Start)
G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).
a(n) = 2^n * A007889(n). (End)

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)

A218308 E.g.f. A(x) satisfies A( x/(exp(4x)cosh(4x)) ) = exp(3x)cosh(3x).

Original entry on oeis.org

1, 3, 42, 1116, 44616, 2394288, 161719200, 13187258304, 1261037553792, 138415816348416, 17155627044653568, 2370099000682257408, 361171910376568571904, 60185513513709805350912, 10887989148395358662270976, 2125192867898778619536457728

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 + 42*x^2/2! + 1116*x^3/3! + 44616*x^4/4! + 2394288*x^5/5! +...
where
A(x) = cosh(3*x) + 3*7^0*cosh(7*x)*x + 3*11^1*cosh(11*x)*x^2/2! + 3*15^2*cosh(15*x)*x^3/3! + 3*19^3*cosh(19*x)*x^4/4! + 3*23^4*cosh(23*x)*x^5/5! +...

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218309, A218310.

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*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*(4*k+3)^(k-1)*cosh((4*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} (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( 3*x - 3/4 * LambertW(-4*x * exp(4*x)) ).
a(n) = 3/2 * Sum_{k=0..n} (4*k+3)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 3/2 * Sum_{k>=0} (4*k+3)^(k-1) * x^k/(1 - (4*k+3)*x)^(k+1). (End)

cp's OEIS Frontend

A218300 E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(2*x)*cosh(2*x).

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A349562 Number of labeled rooted forests with 2-colored leaves.

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

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

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

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

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

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A218300 E.g.f. A(x) satisfies A( x/(exp(x)cosh(x)) ) = exp(2x)cosh(2x).

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).

A218308 E.g.f. A(x) satisfies A( x/(exp(4x)cosh(4x)) ) = exp(3x)cosh(3x).