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A209884
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E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x) * d/dx x^2*A(x)/2.
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2
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1, 2, 10, 88, 1172, 21688, 527576, 16224640, 612742784, 27786137856, 1486036616448, 92365550416896, 6591134271264000, 534423153792751104, 48801114693060804096, 4980221379342136676352, 564172247490745614434304, 70522894891787889987747840
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: exp( Sum_{n>=1} 2*x^n / (n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.
a(n) = (n-1)! * Sum_{k=0..n-1} 2*binomial(n,k)*a(k)/k! for n>0 with a(0)=1.
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 88*x^3/3! + 1172*x^4/4! + 21688*x^5/5! +...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
A(x) + x*A'(x)/2 = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
Also, a(n) appears in the expansion:
B(x) = 1 + 2*x + 10*x^2/2!^2 + 88*x^3/3!^2 + 1172*x^4/4!^2 + 21688*x^5/5!^2 +...
such that
log(B(x)) = 2*x + 2*x^2/(2*2!) + 2*x^3/(3*3!) + 2*x^4/(4*4!) + 2*x^5/(5*5!) +...
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MATHEMATICA
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Table[Sum[BellY[n, k, 2/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
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PROG
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(PARI) {a(n)=local(A=1+x, B); for(i=1, n, B=subst(A, x, x/(1-x+x*O(x^n)))/(1-x); A=1+2*intformal((B-A)/x)); n!*polcoeff(A, n)}
(PARI) {a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 2*binomial(n, k)*a(k)/k!)))}
(PARI) {a(n)=n!^2*polcoeff(exp(sum(m=1, n, 2*x^m/(m*m!))+x*O(x^n)), n)}
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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