cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A130263 Number of degree-n permutations such that number of cycles of size k is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 6, 14, 85, 529, 3451, 26816, 243909, 2507333, 26196841, 323194816, 4086482335, 57669014597, 864137455455, 13792308331616, 231648908415001, 4211676768746569, 79205041816808905, 1584565388341689032, 33265011234209710011, 730971789582886971689
Offset: 0

Views

Author

Vladeta Jovovic, Aug 06 2007

Keywords

Examples

			a(2)=1 because we have (12) ((1)(2) does not qualify). a(4)=14 because the following 10 permutations of 4 do not qualify: (1)(2)(3)(4), (14)(2)(3), (1)(24)(3), (1)(2)(34), (13)(2)(4), (13)(24), (1)(23)(4), (14)(23), (12)(3)(4) and (12)(34).

Links

Crossrefs

Cf. A055922, A130219, A130220, A130268, A218504, A270597, A270598, A270614.

Programs

  • Magma
    m:=40;
f:= func< x | (&*[1 + Sinh(x^j/j): j in [1..m+1]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Mar 18 2023
  • Maple
    g:=product(1+sinh(x^k/k),k=1..40): gser:=series(g,x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..21); # 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)=1, 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(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015
  • Mathematica
    nn = 25; Range[0, nn]!*CoefficientList[Series[Product[1 + Sinh[x^k/k], {k, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 20 2016 *)
  • SageMath
    m=40
def f(x): return product( 1 + sinh(x^j/j) for j in range(1,m+2) )
def A130263_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A130263_list(m) # G. C. Greubel, Mar 18 2023

Formula

E.g.f.: Product_{k>0} (1+sinh(x^k/k)).
a(n) ~ c * n!, where c = A270614 = Product_{k>=1} ((1 + sinh(1/k)) / exp(1/k)) = 0.625635801977949844... . - Vaclav Kotesovec, Mar 20 2016

Extensions

More terms from Emeric Deutsch, Aug 24 2007

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Oct 31 2012

Keywords

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))

Formula

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

Views

Author

Vaclav Kotesovec, Mar 20 2016

Keywords

Links

Crossrefs

Cf. A130263, A130268, A218504, A270597.

Programs

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

Formula

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

Views

Author

Vladeta Jovovic, Aug 06 2007

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Extensions

More terms from Emeric Deutsch, Aug 24 2007

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

Views

Author

Vaclav Kotesovec, Mar 20 2016

Keywords

Examples

			0.625635801977949844597484816103104429905206355112713196896034983259858...

Crossrefs

Cf. A130263, A246945, A249673, A270597.

Programs

  • Maple
    evalf(Product((1+sinh(1/k))/exp(1/k), k=1..infinity), 120);
  • PARI
    default(realprecision, 120); exp(sumpos(k=1, log((1+sinh(1/k))/exp(1/k))))

Formula

Equals limit n->infinity A130263(n)/n!.
Showing 1-5 of 5 results.

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