A052803
Expansion of e.g.f. (-1 + sqrt(1 + 4*log(1-x)))/(2*log(1-x)).
Original entry on oeis.org
1, 1, 5, 44, 566, 9674, 207166, 5343456, 161405016, 5591409720, 218592034584, 9521490534720, 457329182411856, 24014921905589328, 1368772939062117936, 84161443919543331840, 5553011951023694408064, 391360838810043628416384, 29342876851060951124158848
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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spec := [S,{C=Cycle(Z),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
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CoefficientList[Series[-1/(2*Log[1-x]) * (1-(1+4*Log[1-x])^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
A317169
Expansion of e.g.f. BesselI(1,2*log(1 - x))/((1 - x)*log(1 - x)).
Original entry on oeis.org
1, 1, 3, 12, 61, 375, 2699, 22232, 206086, 2122110, 24023623, 296474178, 3960532707, 56931074109, 876098828097, 14369369855760, 250215898045984, 4609913757678432, 89586669708676510, 1831372328505086980, 39284382532454768754, 882269612910279500214, 20703128006754726971507
Offset: 0
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a:=series(BesselI(1,2*log(1 - x))/((1 - x)*log(1 - x)), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[BesselI[1, 2 Log[1 - x]]/((1 - x) Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4], {k, 0, n}], {n, 0, 22}]
A355290
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).
Original entry on oeis.org
1, 1, 1, 0, -3, -2, 23, 17, -333, 86, 6941, -17025, -160267, 1082864, 2273807, -56742606, 152154285, 2293098332, -22007462809, -15179437171, 1671107690083, -10716783889040, -58404948615167, 1439391012463810, -6701658223127029, -88340107011433060
Offset: 0
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A355290 := proc(n)
add((-1)^(n-k)*stirling2(n,k)*A000108(k),k=0..n) ;
end proc:
seq(A355290(n),n=0..70) ; # R. J. Mathar, Mar 13 2023
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a(n) = sum(k=0, n,(-1)^(n-k)*stirling(n, k, 2)*binomial(2*k, k)/(k+1));
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my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(2*k, k)/(k+1)*x^k/prod(j=1, k, 1+j*x)))
A355292
a(n) = Sum_{k=1..n} |Stirling1(n,k)| * Catalan(k-1).
Original entry on oeis.org
1, 2, 7, 34, 208, 1521, 12871, 123306, 1316316, 15471114, 198319614, 2751524557, 41058030388, 655427422651, 11142214939181, 200919300509214, 3829751956014084, 76928721540858772, 1624015067086462504, 35942784684670110710, 832134062464902004336
Offset: 1
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Table[Sum[Abs[StirlingS1[n,k]] * CatalanNumber[k-1], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Jul 01 2022 *)
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a(n) = sum(k=1, n, abs(stirling(n, k, 1))*binomial(2*k-2, k-1)/k);
Showing 1-4 of 4 results.
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