A376041
E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^3)) / (1 - A(x)).
Original entry on oeis.org
0, 1, 9, 191, 6496, 305164, 18317390, 1339293822, 115492112640, 11476262240520, 1291250885222592, 162271449317302632, 22528350072978189600, 3424249337820235241472, 565573503590604522245136, 100864333223422171393303488, 19317041144591537348567168256
Offset: 0
-
a(n) = sum(k=1, n, (3*n+2*k-2)!/(3*n+k-1)!*abs(stirling(n, k, 1)));
A087138
Expansion of (1-sqrt(1-4*log(1+x)))/2.
Original entry on oeis.org
1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360
Offset: 1
-
Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)
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x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017
A371327
E.g.f. satisfies A(x) = -log(1 - x/(1 - A(x)))/(1 - A(x)).
Original entry on oeis.org
0, 1, 5, 59, 1128, 29954, 1019282, 42318296, 2074276320, 117237652008, 7506386360232, 536983774338120, 42447806791009056, 3674351246886880416, 345667310491536157056, 35116581800947400780928, 3831441153568328284066560, 446832269484565155280539264
Offset: 0
-
a(n) = sum(k=1, n, (n+2*k-2)!/(n+k-1)!*abs(stirling(n, k, 1)));
A376042
E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^2)) / (1 - A(x)).
Original entry on oeis.org
0, 1, 7, 116, 3092, 114034, 5378396, 309151968, 20964872624, 1638608258904, 145038615271512, 14340344355439200, 1566483453363376896, 187355848936261332144, 24351019737412176648576, 3417500066845923960657408, 515071814323666902383222784
Offset: 0
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a(n) = sum(k=1, n, (2*n+2*k-2)!/(2*n+k-1)!*abs(stirling(n, k, 1)));
A087152
Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.
Original entry on oeis.org
1, 3, 20, 194, 2554, 42226, 843744, 19769256, 531768120, 16152296424, 546895099200, 20425461026736, 834215500905552, 36988602430554576, 1769524998544143360, 90851799797294235264, 4982968503277896871296
Offset: 1
-
Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/Log[1+x]/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)
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x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/log(1+x)/2 - 1)) \\ G. C. Greubel, May 24 2017
A371314
E.g.f. satisfies A(x) = -log(1 - x)/(1 - A(x))^2.
Original entry on oeis.org
0, 1, 5, 56, 1022, 26054, 853426, 34150584, 1614418536, 88035438144, 5439554576064, 375580703703072, 28658577826251072, 2394815612176027104, 217504341217879448352, 21333409628052488832768, 2247318076016738768083200, 253054488675536428638723840
Offset: 0
-
A371314 := proc(n)
add((3*k-2)!/(2*k-1)!*abs(stirling1(n,k)),k=1..n) ;
end proc:
seq(A371314(n),n=0..40) ; # R. J. Mathar, Mar 25 2024
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Table[Sum[(3*k-2)!/(2*k-1)! * Abs[StirlingS1[n, k]], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 19 2024 *)
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a(n) = sum(k=1, n, (3*k-2)!/(2*k-1)!*abs(stirling(n, k, 1)));
A371315
E.g.f. satisfies A(x) = -log(1 - x)/(1 - A(x))^3.
Original entry on oeis.org
0, 1, 7, 110, 2796, 98754, 4469334, 246741984, 16079405784, 1208082769560, 102810760773096, 9774841791650880, 1026870593449179264, 118121793328191431232, 14766518531481521488704, 1993367920121834019649920, 288988424345833831094150016
Offset: 0
-
a(n) = sum(k=1, n, (4*k-2)!/(3*k-1)!*abs(stirling(n, k, 1)));
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 1-8 of 8 results.