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A121452
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Exponential generating function (1-x^2)^(-1/x).
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7
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1, 1, 1, 4, 13, 71, 391, 2836, 21729, 198829, 1939501, 21515836, 254169301, 3319328299, 45979476635, 691443303916, 10979537304961, 186915474027321, 3345563762493049, 63613875064443796, 1266776073045809341
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OFFSET
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0,4
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LINKS
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FORMULA
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E.g.f.: (1-x^2)^(-1/x) = Sum_{n>=0} a(n)*x^n/n!.
E.g.f.: exp( (1-x) * Sum_{n>=0} x^n * Sum_{k=1..n+1} x^k/k ). - Paul D. Hanna, May 03 2012
a(n) = n!*Sum_{m=floor((n+1)/2)..n} (-1)^(n-m)*Stirling1(m,2*m-n)/m!. - Vladimir Kruchinin, Mar 09 2013
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+1)/2)} (2*k-1)/k * a(n-2*k+1)/(n-2*k+1)!. - Seiichi Manyama, May 01 2022
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EXAMPLE
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E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 71*x^5/5! + 391*x^6/6! + ... such that 1/A(x)^x = 1 - x^2.
The logarithm of the e.g.f. is given by the series:
log(A(x)) = (1-x)*(x + x*(x+x/2) + x^2*(x+x^2/2+x^3/3) + x^3*(x+x^2/2+x^3/3+x^4/4) + x^4*(x+x^2/2+x^3/3+x^4/4+x^5/5) + ...)
log(A(x) = x + x^3/2 + x^5/3 + x^7/4 + x^9/5 + ...
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[(1-x^2)^(-1/x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Sep 28 2013 *)
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PROG
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(PARI) {a(n)=n!*polcoeff((1-x^2 +x^2*O(x^n))^(-1/x), n)}
(PARI) {a(n)=n!*polcoeff(exp((1-x)*sum(m=0, n, x^m*sum(k=1, m+1, x^k/k)+x*O(x^n))), n)} /* Paul D. Hanna, May 03 2012 */
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^2)/x))) \\ Seiichi Manyama, May 01 2022
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+1)\2, (2*j-1)/j*v[i-2*j+2]/(i-2*j+1)!)); v; \\ Seiichi Manyama, May 01 2022
(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, n-2*k, 1))/(n-k)!); \\ Seiichi Manyama, May 01 2022
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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