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
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).
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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
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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
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nn = 25; Range[0, nn]!*CoefficientList[Series[Product[1 + Sinh[x^k/k], {k, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 20 2016 *)
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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
A270664
E.g.f.: Product_{k>=1} (1 + tanh(x^k)).
Original entry on oeis.org
1, 1, 2, 10, 48, 336, 2400, 22240, 220416, 2496256, 30286080, 411725568, 6004838400, 94609106944, 1588301524992, 28577718427648, 546685777182720, 11027370474504192, 234498341381603328, 5253826506085629952, 123695389756163358720, 3039894623920125116416
Offset: 0
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nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[(1+Tanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x]
A270666
E.g.f.: Product_{k>=1} (1 + tan(x^k)).
Original entry on oeis.org
1, 1, 2, 14, 48, 416, 3360, 29504, 274176, 3503104, 45192960, 579956992, 8982251520, 138130720768, 2456648183808, 45868468109312, 871166211686400, 17536583860060160, 393972064172900352, 8704569607311982592, 210657904645299240960, 5322004254737369399296
Offset: 0
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nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[(1+Tan[x^k]), {k, 1, nmax}], {x, 0, nmax}], x]
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
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).
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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 *)
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{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),", "))
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
a(2)=4 because we have (1)(2)(3)(4), (12)(34), (13)(24) and (14)(23).
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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
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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 *)
A270597
E.g.f.: 1/(1-x) * Product_{k>0} (1 + sinh(x^k/k)).
Original entry on oeis.org
1, 2, 5, 21, 98, 575, 3979, 31304, 277248, 2739141, 29898743, 355083014, 4584190984, 63680965127, 949202526375, 15102175351080, 255427113948896, 4573909845546233, 86542053988578763, 1723504067599805402, 36054646740337797072, 790412592781303448523
Offset: 0
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nn=25; Range[0, nn]!*CoefficientList[Series[1/(1-x)*Product[1+Sinh[x^k/k], {k, 1, nn}], {x, 0, nn}], x]
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