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A064856
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Stirling transform of Catalan numbers: a(n) = sum(k=0..n, stirling2(n,k)*binomial(2*k,k)/(k+1) ).
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2
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1, 1, 3, 12, 59, 338, 2185, 15613, 121553, 1020170, 9154963, 87276995, 879242215, 9319182044, 103537712361, 1201967382478, 14540040004755, 182840037042560, 2384985091689409, 32209645344213417, 449608555748234353
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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FORMULA
| O.g.f.: Sum_{n>=1} C(2*n,n)/(n+1) * x^n / Product_{k=0..n} (1-k*x). [From Paul D. Hanna, Jul 20 2011]
E.g.f.: exp(2*exp(z)-2)*(BesselI(0, 2*exp(z)-2)-BesselI(1, 2*exp(z)-2)). Representation as a sum of an infinite series involving the confluent hypergeometric function 1F1, in Maple notation: a(n)=evalf(sum('k'^n*2^(2*'k')*GAMMA('k'+1/2)*evalf(hypergeom(['k'+1/2], ['k'+2], -4))/(sqrt(Pi)*'k'!*('k'+1)!), 'k'=0..infinity)), n=0, 1...
E.g.f.: hypergeom([1/2], [2], 4*(exp(x)-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 11 2003
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MATHEMATICA
| Table[Sum[StirlingS2[n, k] Binomial[2k, k]/(k+1), {k, 0, n}], {n, 0, 20}] (* From Harvey P. Dale, Nov 01 2011 *)
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PROG
| (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/(m!*(m+1)!)*x^m/prod(k=1, m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
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CROSSREFS
| Cf. A000108.
Sequence in context: A105668 A192768 A179325 * A080337 A196710 A196711
Adjacent sequences: A064853 A064854 A064855 * A064857 A064858 A064859
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KEYWORD
| nice,nonn
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AUTHOR
| Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 08 2001
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