A249673 -id:A249673 - cp's OEIS frontend

A218504 E.g.f.: Product_{n>=1} 1/(1 - tanh(x^n/n)).

1, 1, 3, 9, 40, 200, 1286, 9002, 74712, 672408, 6892312, 75815432, 925733216, 12034531808, 170656068480, 2559841027200, 41356302857344, 703057148574848, 12752569691858048, 242298824145302912, 4875886476833445888, 102393616013502363648, 2264106940756915715584

Offset: 0

Paul D. Hanna, Oct 31 2012

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 9*x^3/3! + 40*x^4/4! + 200*x^5/5! +...
where A(x) = 1/((1-tanh(x))*(1-tanh(x^2/2))*(1-tanh(x^3/3))*(1-tanh(x^4/4))*...)
Let G(x) = Product_{n>=1} cosh(x^n/n) be the e.g.f. of A130268:
G(x) = 1 + x^2/2! + 4*x^4/4! + 86*x^6/6! + 2696*x^8/8! + 168232*x^10/10! +...
then e.g.f. A(x) = G(x)/(1-x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..410

Cf. A130263, A130268, A249673, A270597, A270598.

nn = 25; Range[0, nn]! * CoefficientList[Series[1/(1 - x)*Product[Cosh[x^k/k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 20 2016 *)

{a(n)=n!*polcoeff(1/prod(k=1, n, (1-tanh(x^k/k+x*O(x^n)))), n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: 1/(1-x) * Product_{n>=1} cosh(x^n/n); see A130268.

a(n) ~ c * n!, where c = A249673 = Product_{k>=1} cosh(1/k) = 2.1164655365... . - Vaclav Kotesovec, Nov 03 2014

A270598 E.g.f.: Product_{k>0} (1 + tanh(x^k/k)).

Original entry on oeis.org

1, 1, 1, 3, 14, 70, 364, 2548, 21104, 189936, 1830968, 20140648, 241712272, 3142259536, 43472528384, 652087925760, 10396900456448, 176747307759616, 3162885135453952, 60094817573625088, 1198006583353972736, 25158138250433427456, 551506339507727783936

Offset: 0

Vaclav Kotesovec, Mar 20 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A130263, A130268, A218504, A270597.

nn=25; Range[0, nn]! * CoefficientList[Series[Product[1+Tanh[x^k/k], {k, 1, nn}], {x, 0, nn}], x]

E.g.f.: Product_{k>0} exp(x^k/k)/cosh(x^k/k).

a(n) ~ c * n!, where c = 1/A249673 = 1/Product_{k>=1} cosh(1/k) = 0.472485841489...

A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.

Original entry on oeis.org

1, 1, 4, 86, 2696, 168232, 15948032, 2172623168, 398846422144, 97541017510784, 29909993927387648, 11447388459863715328, 5284740632299379566592, 2927671399386587378671616, 1897593132067741963020476416, 1437515129453860805943287939072

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(2)=4 because we have (1)(2)(3)(4), (12)(34), (13)(24) and (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..220

Cf. A055922, A130219, A130220, A130263, A218504, A270597, A270598.

g:=product(cosh(x^k/k),k=1..30): gser:=series(g,x=0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0 or irem(j, 2)=0, multinomial(n, n-i*j, i$j)
       *(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

nn=26;Select[Range[0,nn]!CoefficientList[Series[Product[Cosh[x^k/k],{k,1,nn}],{x,0,nn}],x],#>0&] (* Geoffrey Critzer, Sep 17 2013 *)

E.g.f.: Product_{k>0} cosh(x^k/k).

a(n) ~ c * (2*n-1)! / n ~ c * sqrt(Pi) * n^(2*n-3/2) * 2^(2*n) / exp(2*n), where c = A249673 = Product_{k>=1} cosh(1/k) = 2.1164655365... . - Vaclav Kotesovec, Mar 19 2016

More terms from Emeric Deutsch, Aug 24 2007

A246945 Decimal expansion of the coefficient e^G appearing in the asymptotic expression of the probability that a random n-permutation is a square, as sqrt(2/Pi)*e^G/sqrt(n).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 08 2014

			G = 0.2003084150040401276417752235643787366634879653405876198956293474890714...
e^G = 1.22177951519253683396485298445636121278881014814697728683863962970923...
sqrt(2/Pi)*e^G = 0.974839011877335012323657925154410019528043463671159620094...

See A003483.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 41.
Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO]

Cf. A003483, A249673.

evalf(1/(product(sech(1/(2*k)), k=1..infinity)), 120) # Vaclav Kotesovec, Sep 20 2014

digits = 42; m0 = 10^4; dm = 1000; tail[m_] := (406425600*PolyGamma[1, m] - 2822400*PolyGamma[3, m] + 9408*PolyGamma[5, m] - 17*PolyGamma[7, m])/3251404800; Clear[g]; g[m_] := g[m] = Sum[Log[Cosh[1/(2*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; g[m0] ; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits + 5] != RealDigits[g[m - dm], 10, digits + 5], Print["m = ", m]; m = m + dm]; G = g[m]; RealDigits[E^G, 10, digits ] // First
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[Exp[Sum[(-1)^(n + 1)*Zeta[2*n]^2*(1 - 1/2^(2*n))/n/Pi^(2*n), {n, 1, m}]], 120]], {m, 100, 150}]] (* Vaclav Kotesovec, Sep 20 2014 *)

e^G = prod_{k>=1} cosh(1/(2k)).

G = Sum_{n>=1} (-1)^(n+1) * Zeta(2*n)^2 * (1-1/2^(2*n)) / (n * Pi^(2*n)). - Vaclav Kotesovec, Sep 20 2014

More terms from Vaclav Kotesovec, Sep 20 2014

A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

Original entry on oeis.org

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

Offset: 0

Fredrik Johansson, May 23 2006

			0.38853615333517585918432957568703590501390...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A046990, A046991.
Cf. A051762, A085365, A246945, A230821, A249673.
Cf. A027641, A027642.

nn:= 120:
p:= product(cos(1/n), n=1..infinity):
f:= evalf(p, nn+10):
s:= convert(f, string):
seq(parse(s[n+1]), n=1..nn);  # Alois P. Heinz, Nov 04 2013

S = Series[Log[Cos[x]], {x, 0, 400}]; N[Exp[N[Sum[SeriesCoefficient[S, 2k] Zeta[2k], {k, 1, 200}], 70]], 50]
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[(2^(2*n) - 1)*Zeta[2*n]^2/(n*Pi^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* Vaclav Kotesovec, Sep 20 2014 *)

exp(-sumpos(n=1,-log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; Charles R Greathouse IV, Nov 04 2013

T(n)=((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!
lm=lambertw(2*log(Pi/2)*10^default(realprecision))/2/log(Pi/2); exp(-sum(n=1,lm,T(n)*zeta(2*n))) \\ Charles R Greathouse IV, Nov 06 2013

Equals exp(Sum_{n>=1} -c(n)*zeta(2*n)), where c(n) = A046990(n)/A046991(n).
Equals exp(-Sum_{n>=1} (2^(2*n)-1) * Zeta(2*n)^2 / (n*Pi^(2*n)) ). - Vaclav Kotesovec, Sep 20 2014
Equals exp(Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number. - Amiram Eldar, Jul 30 2023

Corrected offset and extended by Robert G. Wilson v, Nov 03 2013

A270614 Decimal expansion of Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 20 2016

			0.625635801977949844597484816103104429905206355112713196896034983259858...

Cf. A130263, A246945, A249673, A270597.

evalf(Product((1+sinh(1/k))/exp(1/k), k=1..infinity), 120);

default(realprecision, 120); exp(sumpos(k=1, log((1+sinh(1/k))/exp(1/k))))

Equals limit n->infinity A130263(n)/n!.

A363502 Decimal expansion of Product_{k>=1} k*sinh(1/k).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 30 2023

			1.30797093666428364901210447600705632046551568313822...

Similar constants: A118817, A249673, A295219.

Cf. A027641, A027642.

evalf(exp(sum(log(k*sinh(1/k)), k = 1 .. infinity)), 120)

Block[{$MaxExtraPrecision = 1000}, RealDigits[Exp[Sum[(-1)^(k + 1) * Zeta[2*k]^2 / (k*Pi^(2*k)), {k, 1, 200}]], 10, 120][[1]]]

PARI
```
exp(-sumpos(k=1,-log(k*sinh(1/k))))
```

Equals exp(Sum_{k>=1} 2^(2*k-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number.

Equals exp(Sum_{k>=1} (-1)^(k+1)*zeta(2*k)^2/(k*Pi^(2*k))).

cp's OEIS Frontend

A218504 E.g.f.: Product_{n>=1} 1/(1 - tanh(x^n/n)).

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A270598 E.g.f.: Product_{k>0} (1 + tanh(x^k/k)).

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A130268 Number of degree-2n permutations such that number of cycles of size k is even (or zero) for every k.

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A246945 Decimal expansion of the coefficient e^G appearing in the asymptotic expression of the probability that a random n-permutation is a square, as sqrt(2/Pi)*e^G/sqrt(n).

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A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

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A270614 Decimal expansion of Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)).

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A363502 Decimal expansion of Product_{k>=1} k*sinh(1/k).

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