A385750
a(n) = Sum_{k=0..n} Stirling2(n,k) * (n!/k!)^2.
Original entry on oeis.org
1, 1, 5, 64, 1681, 78651, 5891041, 653545390, 101785047169, 21431911982437, 5927319770834701, 2101574777340578156, 935265924020629176625, 512945332353359967175999, 341342159773993944429746793, 272012935493149854994361194426, 256689188247205271953044107166721, 284051735653584424779666013789038985
Offset: 0
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Table[Sum[StirlingS2[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[(Exp[x] - 1)^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3
A227207
E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - n*k*x)).
Original entry on oeis.org
1, 1, 6, 105, 3568, 204745, 18028266, 2278860535, 394667414016, 90302033890953, 26525942216131330, 9775058594870836861, 4433256936788979640848, 2434899483389881601250937, 1597444746833206096334387802, 1237091666097626095124512681755, 1119205949224015886848972396596736
Offset: 0
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
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Flatten[{1,Table[n! * Sum[k^n * StirlingS2[n,k] / k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)
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{a(n)=n!*polcoeff(sum(m=0, 20, m^m*x^m/m!/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))
A336588
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(exp(x) - 1)).
Original entry on oeis.org
1, 2, 9, 67, 725, 10616, 200767, 4740149, 136113217, 4656324934, 186642121061, 8647446227487, 457854954921949, 27435354945248732, 1844986431192663683, 138229607701444447561, 11464234006789370705537, 1046538415206891196153834, 104623195637603009050593697
Offset: 0
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nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!^2
Table[n! Sum[StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]
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a(n) = n! * sum(k=0, n, stirling(n+1,k+1,2) / k!); \\ Michel Marcus, Jul 27 2020
Showing 1-3 of 3 results.