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A221371
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O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2*x + k^2*x^2).
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3
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1, 1, 4, 23, 209, 2744, 49539, 1180281, 35921892, 1360513711, 62770245601, 3466178083312, 225719029475675, 17117740162448105, 1495526385479298140, 149120758170390404103, 16831018302445533666705, 2134813624482300873515304, 302332062412598445891728563
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OFFSET
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0,3
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COMMENTS
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Compare to the o.g.f. of A110501, the unsigned Genocchi numbers (of first kind):
Sum_{n>=0} n!^2 * x^(n+1) / Product_{k=1..n} (1 + k^2*x).
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LINKS
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FORMULA
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O.g.f.: A(x) = 1/(1-x*(1+x)/(1-2*x/(1-4*x*(1+x)/(1-6*x*(1+x)/(1-9*x*(1+x)/(1-12*x*(1+x)/(... -[(n+1)/2]*[(n+2)/2]*x*(1+x)/(1- ...)))))))) (continued fraction).
a(n) = Sum_{k=0..[n/2]} binomial(n-k,k) * A110501(n+1), where A110501(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = Bernoulli numbers).
a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 209*x^4 + 2744*x^5 + 49539*x^6 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!^2*x^2*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)) + 3!^2*x^3*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)) + 4!^2*x^4*(1+x)*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)*(1+16*x+16*x^2)) + ...
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MATHEMATICA
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a[n1_Integer?NonNegative, n2_Integer?NonNegative] := CoefficientList[Sum[(m!)^2*x^m*Product[(1 + x)/(1 + k^2*x + k^2*x^2), {k, 1, m}], {m, 0, n2 + 1}] + O[x]^(n2 + 2), x][[n1 + 1 ;; n2 + 1]]; a[0, 18] (* Robert P. P. McKone, Sep 16 2023 *)
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PROG
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(PARI) {a(n)=polcoeff( sum(m=0, n, m!^2*x^m*prod(k=1, m, (1+x)/(1+k^2*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n\2, binomial(n-k, k)*2*(-1)^(n-k+1)*(1-4^(n-k+1))*bernfrac(2*(n-k+1)))}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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