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
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),", "))
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
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nn=25; Range[0, nn]! * CoefficientList[Series[Product[1+Tanh[x^k/k], {k, 1, nn}], {x, 0, nn}], x]
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]
A374262
Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 316, 3564, 27756, 141516, 556656, 6678816, 73015944, 521124696, 6144018336, 75200767776, 677927254176, 4642387894944, 75217104395136, 1167068528384256, 12348761954020416, 97377968145352896, 882819252604721664, 66882151986021043200
Offset: 0
a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(combinat[multinomial](n, i$i*j, n-i^2*j)*
b(n-i^2*j, i-1)*(i-1)!^(i*j)/(i*j)!, j=0..n/i^2))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
Showing 1-5 of 5 results.