A007840 -id:A007840 - cp's OEIS frontend

A079640 Matrix product of unsigned Stirling1-triangle |A008275(n,k)| and unsigned Lah-triangle |A008297(n,k)|.

1, 3, 1, 14, 9, 1, 88, 83, 18, 1, 694, 860, 275, 30, 1, 6578, 10084, 4245, 685, 45, 1, 72792, 132888, 69244, 14735, 1435, 63, 1, 920904, 1950024, 1209880, 318969, 41020, 2674, 84, 1, 13109088, 31580472, 22715972, 7133784, 1137549, 98028, 4578, 108, 1

Offset: 1

Vladeta Jovovic, Jan 30 2003

Also the Bell transform of A007840(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			1; 3,1; 14,9,1; 88,83,18,1; 694,860,275,30,1; 6578,10084,4245,685,45,1; ...

Cf. A007840 (first column).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(k!*abs(combinat:-stirling1(n+1, k)), k=0..n+1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, Sum[k!*Abs[StirlingS1[n+1, k]], {k, 0, n+1}]], rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} |A008275(n, i)| * |A008297(i, k)|.

E.g.f.: (1-x)^(-y/(1+log(1-x))). - Vladeta Jovovic, Nov 22 2003

A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 9, 3, 1, 88, 56, 18, 4, 1, 694, 440, 140, 30, 5, 1, 6578, 4164, 1320, 280, 45, 6, 1, 72792, 46046, 14574, 3080, 490, 63, 7, 1, 920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1, 13109088, 8288136, 2620512, 552552, 87444, 11088, 1176, 108, 9, 1

Offset: 0

Peter Luschny, Sep 15 2017

			Triangle starts:
[1]
[1,           1]
[3,           2,      1]
[14,          9,      3,     1]
[88,         56,     18,     4,    1]
[694,       440,    140,    30,    5,   1]
[6578,     4164,   1320,   280,   45,   6,  1]
[72792,   46046,  14574,  3080,  490,  63,  7, 1]
[920904, 582336, 184184, 38864, 6160, 784, 84, 8, 1]

Row sums: A291978.
Columns: A007840 (c=1), A052860 (c=2).
Diagonal: A045943 (d=3).
Cf. A291980.

T_row := proc(n) exp(x*t)/(1 + log(1+x)): series(%, x, n+1):
seq(coeff((-1)^(n-k)*n!*coeff(%,x,n),t,k), k=0..n) end:
seq(T_row(n), n=0..9);

T[n_, k_] := Binomial[n, n - k]*Sum[j!*Abs[StirlingS1[n - k, j]], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)

T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*abs(Stirling1(n - k, j)). - Detlef Meya, May 12 2024

A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

Original entry on oeis.org

1, 1, 5, 32, 274, 2939, 37833, 568210, 9753280, 188342949, 4041170695, 95380234366, 2455830637412, 68501591450447, 2057726452045145, 66227424015265178, 2273614433910697920, 82932491842062712873, 3202994529476330549163, 130577628147690206429038, 5603479009890212632226756

Offset: 0

Ilya Gutkovskiy, May 31 2018

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 32*x^3/3! + 274*x^4/4! + 2939*x^5/5! + 37833*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..399
Index entries for sequences related to logarithmic numbers

Cf. A002104, A006153, A007840, A009324, A191365.

a:=series(1/(1+log(1-x)*exp(x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[1/(1 + Log[1 - x] Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{1, 1, 1 - k}, {2}, -1] a[n - k]/(k - 1)!, {k, 1, n}]; Table[n! a[n], {n, 0, 20}]

a(n) ~ n! / ((1 + exp(r)/r) * (1 - exp(-r))^(n+1)), where r = 0.62747017959751658496114808922921433658821962606026068561095... is the root of the equation r*exp(1 - exp(-r)) = 1. - Vaclav Kotesovec, Mar 26 2019

a(0) = 1; a(n) = Sum_{k=1..n} A002104(k) * binomial(n,k) * a(n-k). - Seiichi Manyama, May 04 2022

A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 1, 8, 114, 2358, 64074, 2157828, 86714592, 4049302404, 215458069428, 12867377875632, 852254389954296, 61998666080311800, 4914000741835488744, 421488717980664846960, 38897664480760253351904, 3843081247426270376211216, 404727487161912602921083536

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

In general, for k >= 1, if e.g.f. = 1 / (1 + k*log(1 - x))^(1/k), then a(n) ~ n! * exp(n/k) / (Gamma(1/k) * k^(1/k) * n^(1 - 1/k) * (exp(1/k) - 1)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A007840, A008542, A346978, A346985, A346987, A347015, A347016.

nmax = 17; CoefficientList[Series[1/(1 + 6 Log[1 - x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A008542(k).

a(n) ~ n! * exp(n/6) / (Gamma(1/6) * 6^(1/6) * n^(5/6) * (exp(1/6) - 1)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021

A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

Original entry on oeis.org

1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136

Offset: 0

Ilya Gutkovskiy, Mar 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..233

Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.

Cf. A081048.

Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]

a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022

a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.

a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - Vaclav Kotesovec, Mar 03 2022

A354244 Expansion of e.g.f. Sum_{k>=0} (2k)! (-log(1-x))^k / k!.

Original entry on oeis.org

1, 2, 26, 796, 44916, 4058448, 537029616, 97903213056, 23525415709632, 7205450503530816, 2740066802232081984, 1266655419369548369280, 699532666466320784246400, 454880976674201215672273920, 344008843780994236543882521600

Offset: 0

Seiichi Manyama, May 20 2022

nonn

Cf. A007840, A354251.

Cf. A316747, A354241.

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k)!*(-log(1-x))^k/k!)))

a(n) = sum(k=0, n, (2*k)!*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (2*k)! * |Stirling1(n,k)|.

A355652 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 3, 32, 694, 1, 0, 0, 0, 6, 150, 6578, 1, 0, 0, 0, 4, 20, 1524, 72792, 1, 0, 0, 0, 0, 10, 270, 12600, 920904, 1, 0, 0, 0, 0, 5, 40, 1764, 147328, 13109088, 1, 0, 0, 0, 0, 0, 15, 210, 12600, 1705536, 207360912, 1, 0, 0, 0, 0, 0, 6, 70, 2464, 146880, 23681520, 3608233056

Offset: 0

Seiichi Manyama, Jul 13 2022

			Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    14,    3,   3,  0,  0, 0, 0, ...
    88,   32,   6,  4,  0, 0, 0, ...
   694,  150,  20, 10,  5, 0, 0, ...
  6578, 1524, 270, 40, 15, 6, 0, ...

Amiram Eldar, Antidiagonals n = 0..150, flattened

Columns k=0..3 give A007840, A052830, A351505, A351506.

Cf. A355610, A355665.

T[n_, k_] := n! * Sum[j! * Abs[StirlingS1[n - k*j, j]]/(k!^j*(n - k*j)!), {j, 0, Floor[n/(k + 1)]}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)

T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));

T(0,k) = 1 and T(n,k) = (n!/k!) * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(k!^j * (n-k*j)!).

A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 14, 0, 1, 0, 0, 6, 88, 0, 1, 0, 0, 6, 46, 694, 0, 1, 0, 0, 0, 36, 340, 6578, 0, 1, 0, 0, 0, 24, 210, 3308, 72792, 0, 1, 0, 0, 0, 0, 240, 2070, 36288, 920904, 0, 1, 0, 0, 0, 0, 120, 2040, 24864, 460752, 13109088, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 310632, 6551424, 207360912, 0

Offset: 0

Seiichi Manyama, Oct 18 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   3,   2,   0,   0,   0, ...
  0,  14,   6,   6,   0,   0, ...
  0,  88,  46,  36,  24,   0, ...
  0, 694, 340, 210, 240, 120, ...

Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.

Cf. A357119, A357868, A357882.

T(n, k) = sum(j=0, n, (k*j)!*abs(stirling(n, k*j, 1)));

T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(-log(1-x+x*O(x^n)))^k), n));

For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).

A375798 Expansion of e.g.f. 1/(1 + (log(1 - x^2))/x).

Original entry on oeis.org

1, 1, 2, 9, 48, 340, 2820, 27720, 309120, 3897936, 54472320, 838918080, 14080651200, 256214724480, 5018771197440, 105361754097600, 2358985057228800, 56124276848640000, 1413738138502609920, 37591686093776855040, 1052149579611011481600

Offset: 0

Seiichi Manyama, Aug 29 2024

nonn

Cf. A007840, A375799.

Cf. A121452.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x^2)/x)))

a(n) = n!*sum(k=0, n\2, (n-2*k)!*abs(stirling(n-k, n-2*k, 1))/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)! * |Stirling1(n-k,n-2*k)|/(n-k)!.

A375799 Expansion of e.g.f. 1/(1 + (log(1 - x^3))/x^2).

Original entry on oeis.org

1, 1, 2, 6, 36, 240, 1800, 16800, 178080, 2086560, 27518400, 399168000, 6286896000, 107623676160, 1984274772480, 39143052748800, 824445099878400, 18450791322163200, 437015358530150400, 10929450232744243200, 287728555881102336000, 7952251084537503744000

Offset: 0

Seiichi Manyama, Aug 29 2024

nonn

Cf. A007840, A375798.

Cf. A353223.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x^3)/x^2)))

a(n) = n!*sum(k=0, n\3, (n-3*k)!*abs(stirling(n-2*k, n-3*k, 1))/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)! * |Stirling1(n-2*k,n-3*k)|/(n-2*k)!.

cp's OEIS Frontend

A079640 Matrix product of unsigned Stirling1-triangle |A008275(n,k)| and unsigned Lah-triangle |A008297(n,k)|.

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A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)*n!*[t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

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A305407 Expansion of e.g.f. 1/(1 + log(1 - x)*exp(x)).

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A347019 E.g.f.: 1 / (1 + 6 * log(1 - x))^(1/6).

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A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

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A354244 Expansion of e.g.f. Sum_{k>=0} (2*k)! * (-log(1-x))^k / k!.

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A355652 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).

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A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.

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A291978 Triangle read by rows, T(n, k) = (-1)^(n-k)n![t^k]([x^n] exp(x*t)/(1 + log(1+x))) for 0<=k<=n.

A354244 Expansion of e.g.f. Sum_{k>=0} (2k)! (-log(1-x))^k / k!.

A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! |Stirling1(n,k*j)|.