A130220 -id:A130220 - cp's OEIS frontend

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

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

Vladeta Jovovic, Aug 06 2007

			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).

Alois P. Heinz, Table of n, a(n) for n = 0..450

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

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

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

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

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

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

More terms from Emeric Deutsch, Aug 24 2007

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

A270669 E.g.f.: Product_{k>=1} (1 + sinh(k*x^k)).

Original entry on oeis.org

1, 1, 4, 31, 168, 1841, 19320, 226885, 2655408, 47569681, 743996880, 12582916061, 245804712120, 4831304993113, 109782586920552, 2669560767444901, 61579705719702240, 1566459883903878305, 44585240861695115808, 1212424119941953292461, 37517727808419084095400

Offset: 0

Vaclav Kotesovec, Mar 21 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A082579, A130220, A130263, A255807, A270294, A270670.

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

A270670 E.g.f.: Product_{k>=1} (1 + k*sinh(x^k)).

Original entry on oeis.org

1, 1, 4, 31, 168, 1841, 18600, 221845, 2655408, 44969041, 703172880, 11894018621, 231354830520, 4504644624793, 100890401218152, 2370351246834901, 55456622199548640, 1400307612079837985, 39002429830457675808, 1058964187034314179181, 32049467535091477285800

Offset: 0

Vaclav Kotesovec, Mar 21 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..432

Cf. A130220, A130263, A270294, A270669.

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

cp's OEIS Frontend

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

Mathematica

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A270669 E.g.f.: Product_{k>=1} (1 + sinh(k*x^k)).

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Mathematica

A270670 E.g.f.: Product_{k>=1} (1 + k*sinh(x^k)).

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Mathematica