A073596
Expansion of e.g.f. exp(x) * log(1-x)/(x-1).
Original entry on oeis.org
0, 1, 5, 23, 116, 669, 4429, 33375, 283072, 2673321, 27845293, 317274407, 3926774180, 52469606981, 752922837861, 11549166072847, 188596608142560, 3266826328953745, 59830416584102325, 1155208913864163511, 23453274942011893556, 499481183766226468013
Offset: 0
Column k=2 of
A269951 (with a different offset).
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*Log(1-x)/(x-1))); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Aug 28 2018
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b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:
a:= proc(n) add(b(k)*binomial(n, k), k=0..n) end:
seq(a(n), n=0..25); # Alois P. Heinz, Mar 07 2018
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nn=19;Range[0,nn]!CoefficientList[Series[Exp[x]Log[1/(1-x)]/(1-x),{x,0,nn}],x] (* Geoffrey Critzer, Sep 24 2013 *)
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x='x+O('x^30); concat([0], Vec(serlaplace(exp(x)*log(1-x)/(x-1)))) \\ G. C. Greubel, Aug 28 2018
A269954
Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1
Offset: 0
Triangle starts:
1;
0, 1;
0, 0, 1;
0, 1, 1, 1;
0, 2, 5, 3, 1;
0, 9, 20, 17, 6, 1;
0, 44, 109, 100, 45, 10, 1;
0, 265, 689, 694, 355, 100, 15, 1;
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A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
seq(seq(A269954(n, k), k=0..n), n=0..9);
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Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]
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T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025
A381024
Expansion of e.g.f. log(1-x)^2 * exp(x) / (2 * (1-x)).
Original entry on oeis.org
0, 0, 1, 9, 65, 470, 3634, 30681, 284066, 2878284, 31777851, 380396665, 4912874691, 68142259874, 1010736134108, 15970709345353, 267890182932228, 4755088551397016, 89059375695649173, 1755426336571939497, 36327033843657558661, 787539492771039394158, 17850021806783323801766
Offset: 0
Column k=3 of
A269951 (with a different offset).
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nmax=22;CoefficientList[Series[Log[1-x]^2* Exp[x]/ (2* (1-x)),{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 12 2025 *)
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a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k+1, 3, 1)));
A269952
Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1
Offset: 0
1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.
Variant:
A143494 (the main entry for this triangle).
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A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);
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Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]
A381025
Expansion of e.g.f. -log(1-x)^3 * exp(x) / (6 * (1-x)).
Original entry on oeis.org
0, 0, 0, 1, 14, 145, 1415, 14084, 147532, 1646714, 19664350, 251282911, 3430766658, 49928212971, 772465487885, 12671188958674, 219793939324536, 4021442067435092, 77425990864146652, 1565193235764750557, 33153390461212914806, 734397759275046673253, 16982466756411641668051
Offset: 0
Column k=4 of
A269951 (with a different offset).
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nmax=22; CoefficientList[Series[-Log[1-x]^3*Exp[x]/(6*(1-x)),{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 12 2025 *)
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a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k+1, 4, 1)));
Showing 1-5 of 5 results.
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