A003713 -id:A003713 - cp's OEIS frontend

A111933 Triangle read by rows, generated from Stirling cycle numbers.

1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 15, 35, 24, 1, 5, 26, 105, 228, 120, 1, 6, 40, 234, 947, 1834, 720, 1, 7, 57, 440, 2696, 10472, 17582, 5040, 1, 8, 77, 741, 6170, 37919, 137337, 195866, 40320, 1, 9, 100, 1155, 12244, 105315, 630521, 2085605, 2487832, 362880

Offset: 1

Gary W. Adamson, Aug 21 2005

Let M = the infinite lower triangular matrix of Stirling cycle numbers (A008275). Perform M^n * [1, 0, 0, 0, ...] forming an array. Antidiagonals of that array become the rows of this triangle.

			Row 5 of the triangle = 1, 4, 15, 35, 24; generated from M^n * [1,0,0,0,...] (n = 1 through 5); then take antidiagonals.
Terms in the array, first few rows are:
  1, 1,  2,   6,    24,    120, ...
  1, 2,  7,  35,   228,   1834, ...
  1, 3, 15, 105,   947,  10472, ...
  1, 4, 26, 234,  2697,  37919, ...
  1, 5, 40, 440,  6170, 105315, ...
  1, 6, 57, 741, 12244, 245755, ...
  ...
First few rows of the triangle are:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  7,   6;
  1, 4, 15,  35,  24;
  1, 5, 26, 105, 228,  120;
  1, 6, 40, 234, 947, 1834, 720;
  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Row 1..7 give A000142(n-1), A003713, A000268, A000310, A000359, A000406, A001765.
Column 3 of the array = A005449.
Column 4 of the array = A094952.
Cf. A008275, A302358.

a(28), a(36) and a(45) corrected by Seiichi Manyama, Feb 11 2022

A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 7, -6, 1, 0, -35, 40, -12, 1, 0, 228, -315, 130, -20, 1, 0, -1834, 2908, -1485, 320, -30, 1, 0, 17582, -30989, 18508, -5005, 665, -42, 1, 0, -195866, 375611, -253400, 81088, -13650, 1232, -56, 1, 0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1

Offset: 0

Peter Luschny, Jun 27 2019

			Triangle starts:
[0] [1]
[1] [0,       1]
[2] [0,      -2,        1]
[3] [0,       7,       -6,       1]
[4] [0,     -35,       40,     -12,        1]
[5] [0,     228,     -315,     130,      -20,      1]
[6] [0,   -1834,     2908,   -1485,      320,    -30,      1]
[7] [0,   17582,   -30989,   18508,    -5005,    665,    -42,    1]
[8] [0, -195866,   375611, -253400,    81088, -13650,   1232,  -56,   1]
[9] [0, 2487832, -5112570, 3805723, -1389612, 279048, -32130, 2100, -72, 1]

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 8.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Columns k=0..3 give A000007, (-1)^(n+1) * A003713(n), (-1)^n * A341587(n), (-1)^(n+1) * A341588(n).

Cf. A039814 (variant), A129062, A325873.

p[n_] := Sum[StirlingS1[n, k] FactorialPower[x, k] , {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ Seiichi Manyama, Apr 18 2025

def a_row(n):
    s = sum((-1)^(n-k)*stirling_number1(n,k)*falling_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

From Seiichi Manyama, Apr 18 2025: (Start)
T(n,k) = Sum_{j=k..n} Stirling1(n,j) * Stirling1(j,k).
E.g.f. of column k (with leading zeros): f(x)^k / k! with f(x) = log(1 + log(1 + x)). (End)

A336438 a(n) = (n!)^n * [x^n] -log(1 - Sum_{k>=1} x^k / k^n).

Original entry on oeis.org

0, 1, 3, 107, 109720, 5916402624, 25690641168448256, 12501662072725447325457536, 901886074956174349048867091963183104, 12343856662712388173832816538241443833756015132672, 39989244654801819205752864236178211163455535276138236680981184512

Offset: 0

Ilya Gutkovskiy, Jul 21 2020

nonn

Cf. A003713, A074708, A336436, A336437, A336439, A336440.

Table[(n!)^n SeriesCoefficient[-Log[1 - Sum[x^k/k^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 0, ((n - 1)!)^k + (1/n) Sum[(Binomial[n, j] (n - j - 1)!)^k j b[j, k], {j, 1, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]

A346966 Expansion of e.g.f. -log( 1 - log(1 - x)^2 / 2 ).

Original entry on oeis.org

1, 3, 14, 80, 559, 4599, 43665, 470196, 5666586, 75600690, 1106587008, 17636532264, 304092954138, 5640892517610, 112029356591862, 2371963759970352, 53338181764577304, 1269586152655203672, 31891196481381667008, 843109673024218773600, 23400930987874505081160

Offset: 2

Ilya Gutkovskiy, Aug 09 2021

nonn

Cf. A000254, A003713, A346921, A346944.

nmax = 22; CoefficientList[Series[-Log[1 - Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = Abs[StirlingS1[n, 2]] + (1/n) Sum[Binomial[n, k] Abs[StirlingS1[n - k, 2]] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 22}]

a(n) = |Stirling1(n,2)| + (1/n) * Sum_{k=1..n-1} binomial(n,k) * |Stirling1(n-k,2)| * k * a(k).
a(n) ~ (n-1)! / (1 - exp(-sqrt(2)))^n. - Vaclav Kotesovec, Jun 04 2022
a(n) = Sum_{k=1..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A373856 a(n) = Sum_{k=1..n} k! * k^(2n-1) |Stirling1(n,k)|.

Original entry on oeis.org

0, 1, 17, 1652, 474770, 301474214, 357901156354, 712632435944568, 2204970751341231816, 10017874331177386762512, 63973486554110386836270096, 554598491512901862814742673168, 6344773703149123365957506715989568, 93563015826037060521986513216617599504

Offset: 0

Seiichi Manyama, Jun 19 2024

nonn

Cf. A003713, A373855.
Cf. A220179, A242228, A373858.
Cf. A351136, A373861.

nmax=13; Range[0,nmax]!CoefficientList[Series[Sum[(-Log[1 - k^2*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)

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

E.g.f.: Sum_{k>=1} (-log(1 - k^2*x))^k / k.

A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 11, 12, 6, 24, 50, 70, 60, 24, 120, 274, 450, 510, 360, 120, 720, 1764, 3248, 4410, 4200, 2520, 720, 5040, 13068, 26264, 40614, 47040, 38640, 20160, 5040, 40320, 109584, 236248, 403704, 538776, 544320, 393120, 181440, 40320

Offset: 1

N. J. A. Sloane, Apr 14 2011

Also the coefficients of the polynomials which are generated by the exponential generating function -log(1 + x*log(1 - t)). The polynomials might be called 'logarithmic polynomials'. Note also A003713, and A263634 for a different use of this term. See the paper of F. Qi for a related, but different family of polynomials. - Peter Luschny, Jul 11 2020
Edgar remarks that these coefficients are related to Stirling numbers of the second kind (cf. A008277).
The first column and the main diagonal are the factorials (A000142). The n-th entry on the first subdiagonal is A001710(n+1). The second column is A000254, the third column is 2*A000399, and the fourth column is 6*A000454. In general, the k-th column is (k-1)!*s(n,k), where s(n,k) is the unsigned Stirling number of the first kind. - Nathaniel Johnston, Apr 15 2011
With offset n=0, k=0 : triangle T(n,k), read by rows,given by T(n,k) = k*T(n-1, k-1) + n*T(n-1, k) with T(0, 0) = 1. - Philippe Deléham, Oct 04 2011

			Triangle begins:
1
1    1
2    3    2
6    11   12   6
24   50   70   60   24
120  274  450  510  360  120
...

Nathaniel Johnston, Table of n, a(n) for n = 1..2500
G. A. Edgar, Transseries for beginners, arXiv:0801.4877 [math.RA], 2008-2009.
F. Qi, On multivariate logarithmic polynomials and their properties, Indagationes Mathematicae (2018).
Wikipedia, Stirling numbers of the first kind

Cf. A277408, A003713, A263634.

S:=proc(n,k)global s:if(n=0 and k=0)then s[0,0]:=1:elif(n=0 or k=0)then s[n,k]:=0:elif(not type(s[n,k],integer))then s[n,k]:=(n-1)*S(n-1,k)+S(n-1,k-1):fi:return s[n,k]:end:
T:=proc(n,k)return (k-1)!*S(n,k);end:
for n from 1 to 6 do for k from 1 to n do print(T(n,k)):od:od: # Nathaniel Johnston, Apr 15 2011
# With offset n = 0, k = 0:
A188881 := (n, k) -> k!*abs(Stirling1(n+1, k+1)):
seq(seq(A188881(n,k), k=0..n), n=0..8); # Peter Luschny, Oct 19 2017
# Alternative:
gf := -log(1 + x*log(1 - t)): ser := series(gf, t, 18):
toeff := n -> n!*expand(coeff(ser, t, n)):
seq(print(seq(coeff(toeff(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Jul 10 2020

Table[(k - 1)! * Sum[StirlingS2[i, k] * (-1)^(n - i) * StirlingS1[n, i], {i, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Apr 17 2015 *)

T(n,k):=(k-1)!*sum(stirling2(i,k)*(-1)^(n-i)*stirling1(n,i),i,0,k); /* Vladimir Kruchinin, Apr 17 2015 */

{T(n, k) = if( k<1 || k>n, 0, (n-1)! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^n, n-k))}; /* Michael Somos, May 10 2017 */

{T(n, k) = if( k<1 || n<0, 0, (k-1)! * sum(i=0, k, stirling(i, k, 2) * (-1)^(n-i) * stirling(n, i, 1)))}; /* Michael Somos, May 10 2017 */

T(n, k) = (k-1)!*Sum_{i=0..k}(Stirling2(i,k)*(-1)^(n-i)*Stirling1(n,i)) =
T(n, k) = Sum_{i=0..k}(W(i,k)*(-1)^(n-i)*Stirling1(n,i)), where W(n,k) is the Worpitzky triangle A028246. - Vladimir Kruchinin, Apr 17 2015.
T(n,k) = [x^k] n!*[t^n](-log(1 + x*log(1 - t))). - Peter Luschny, Jul 10 2020
T(n,k) = Sum_{m=0..n-k} abs(Stirling1(n-1,m+k-1))*(k+m-1)!/m!. - Vladimir Kruchinin, Jul 14 2025

a(11)-a(45) from Nathaniel Johnston, Apr 15 2011

A277408 Triangle, read by rows, where the g.f. of row n equals the sum of permutations of compositions of functions (1 + kyx) for k=1..n with parameter y independent of variable x, as evaluated at x=1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 12, 22, 36, 24, 60, 140, 300, 576, 120, 360, 1020, 2700, 6576, 14400, 720, 2520, 8400, 26460, 77952, 211680, 518400, 5040, 20160, 77280, 282240, 974736, 3151680, 9408960, 25401600, 40320, 181440, 786240, 3265920, 12930624, 48444480, 170098560, 552303360, 1625702400, 362880, 1814400, 8769600, 40824000, 182226240, 775656000, 3126297600, 11820816000, 41391544320, 131681894400, 3628800, 19958400, 106444800, 548856000, 2726317440, 12989592000, 59044550400, 254303280000, 1028448368640, 3856920883200, 13168189440000, 39916800, 239500800, 1397088000, 7903526400, 43233886080, 227885011200, 1152535824000, 5563643500800, 25464033745920, 109530230261760, 437429486592000, 1593350922240000

Offset: 0

Paul D. Hanna, Oct 16 2016

			Illustration of initial row polynomials.
  R_0(y) = 1;
  R_1(y) = 1 + y;
  R_2(y) = 2 + 3*y + 4*y^2;
  R_3(y) = 6 + 12*y + 22*y^2 + 36*y^3;
  R_4(y) = 24 + 60*y + 140*y^2 + 300*y^3 + 576*y^4;
  R_5(y) = 120 + 360*y + 1020*y^2 + 2700*y^3 + 6576*y^4 + 14400*y^5;
  R_6(y) = 720 + 2520*y + 8400*y^2 + 26460*y^3 + 77952*y^4 + 211680*y^5 + 518400*y^6;
  R_7(y) = 5040 + 20160*y + 77280*y^2 + 282240*y^3 + 974736*y^4 + 3151680*y^5 + 9408960*y^6 + 25401600*y^7;
  ...
Generating method.
  R_0(y) = 1, by convention;
  R_1(y) = Sum_{i=1..1} (1 + i*y);
  R_2(y) = Sum_{i=1..2, j=1..2, j<>i} (1 + i*y*(1 + j*y));
  R_3(y) = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*y*(1 + j*y*(1 + k*y)));
  R_4(y) = Sum_{i=1..4, j=1..4, k=1..4, m=1..4, i,j,k,m distinct} (1 + i*y*(1 + j*y*(1 + k*y*(1 + m*y))));
  etc.
This triangle of coefficients begins:
  1;
  1, 1;
  2, 3, 4;
  6, 12, 22, 36;
  24, 60, 140, 300, 576;
  120, 360, 1020, 2700, 6576, 14400;
  720, 2520, 8400, 26460, 77952, 211680, 518400;
  5040, 20160, 77280, 282240, 974736, 3151680, 9408960, 25401600;
  40320, 181440, 786240, 3265920, 12930624, 48444480, 170098560, 552303360, 1625702400;  ...

Qiaochu Yuan, Has anyone seen this combinatorial identity involving the Bernoulli and Stirling numbers, Math StackExchange, Apr 05 2015.

Cf. A277406 (row sums), A277405, A277407, A188881, A003713.

{T(n,k) = k!*(n-k)! * sum(i=0,n-k+1, (-1)^(n-i+1) * stirling(i,n-k+1,2) * stirling(n+1,i,1))}
for(n=0,11,for(k=0,n,print1( T(n,k) ,", "));print(""))

{T(n, k) = if( k<0 || k>n, 0, n! * k! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^(n+1), k))}; /* Michael Somos, May 10 2017 */

T(n,k) = k!*(n-k)! * Sum_{i=0..n-k+1} (-1)^(n-i+1) * Stirling2(i,n-k+1) * Stirling1(n+1,i). [From formula in A188881 by Vladimir Kruchinin]
T(n,k) = k! * A188881(n+1, n-k+1).
A003713(n) = Sum_{k=0..n} T(n,k) / k!, where e.g.f. of A003713 is log(1/(1+log(1-x))).
Row sums yield A277406.

A330498 Expansion of e.g.f. Sum_{k>=1} log(1 + log(1/(1 - x))^k) / k.

Original entry on oeis.org

0, 1, 1, 6, 24, 152, 1230, 12646, 141274, 1730984, 23920800, 379364664, 6766026168, 131337466608, 2713274041296, 59397879195456, 1386647548658496, 34745321580075648, 934708252265232768, 26835517455387452928, 815158892950448937984

Offset: 0

Vaclav Kotesovec, Dec 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A003713, A330352, A330493, A330499.

nmax = 20; CoefficientList[Series[Sum[Log[1+Log[1/(1-x)]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * c / (n * (1 - exp(-1))^n), where c = 0.6931..., conjecture: c = log(2).

A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.

Original entry on oeis.org

1, 2, 40, 1485, 81088, 5856900, 526685269, 56704848200, 7112345477952, 1018548226480356, 163987811350464660, 29321558852248050388, 5764958268855541178967, 1236150756215397667568170, 287086392921014590422630300, 71789589754855255636302048525, 19231403740347427723119910379040

Offset: 0

Ilya Gutkovskiy, Feb 15 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A003713, A008275, A039814, A321712, A341587, A341588.

Table[Sum[Abs[StirlingS1[2 n, k] StirlingS1[k, n]], {k, n, 2 n}], {n, 0, 16}]
Table[((2 n)!/n!) SeriesCoefficient[(-Log[1 + Log[1 - x]])^n, {x, 0, 2 n}], {n, 0, 16}]

a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1)*stirling(k, n, 1))); \\ Michel Marcus, Feb 16 2021

a(n) = ((2*n)!/n!) * [x^(2*n)] (-log(1 + log(1 - x)))^n.
From Vaclav Kotesovec, Feb 15 2021: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = -16*p*q^2 * log(-2*q/(1+r))^(1+r) / ((1 + 2*q + r)^2 * (1 + 1/(p*(1+r)))^r) = 17.84101281316291323354184111891200669611476053165484517795417711039479218...
p = LambertW(-1, -1/(exp(1/(1+r))*(1+r)))
q = LambertW(-1, -(1+r)/exp((1+r)/2)/2)
r = 0.5094050884976689299791685259225203723646676600942448390861428232759777841...
is the root of the equation (1+p)*(1+r)^2 * (1 + 2*q + r) * log(-p*(1+r)) + 2*log(-(1+r)/(2*q)) * ((1+q)*(1 + p + p*r) - (1+r) * log(-p*(1+r)) * (p - q + r + p*r + (1+p) * (1+q) * (1+r) * (log(1 + 1/(p*(1+r))) - log(-log(-(1+r)/(2*q)))))) = 0
and c = 0.1417076025518808268972093339771762801784527709... (End)

A383170 Expansion of e.g.f. -log(1 + log(1 - 2*x)/2).

Original entry on oeis.org

0, 1, 3, 16, 122, 1208, 14704, 212336, 3547984, 67337728, 1430990976, 33664165632, 868592478720, 24390846882816, 740570519159808, 24177326011834368, 844599686386919424, 31438092340685144064, 1242230898248798896128, 51933512200489564962816, 2290351520336982559358976

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Cf. A383171, A383172.

Cf. A003713, A227917.

a(n) = sum(k=1, n, 2^(n-k)*abs(stirling(n, k, 1)*stirling(k, 1, 1)));

a(n) = Sum_{k=1..n} 2^(n-k) * |Stirling1(n,k) * Stirling1(k,1)|.

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/2) / (exp(1) - exp(-1))^n. - Vaclav Kotesovec, Apr 18 2025

cp's OEIS Frontend

A111933 Triangle read by rows, generated from Stirling cycle numbers.

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A325872 T(n, k) = [x^k] Sum_{k=0..n} Stirling1(n, k)*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A336438 a(n) = (n!)^n * [x^n] -log(1 - Sum_{k>=1} x^k / k^n).

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A346966 Expansion of e.g.f. -log( 1 - log(1 - x)^2 / 2 ).

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A373856 a(n) = Sum_{k=1..n} k! * k^(2*n-1) * |Stirling1(n,k)|.

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A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

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A277408 Triangle, read by rows, where the g.f. of row n equals the sum of permutations of compositions of functions (1 + k*y*x) for k=1..n with parameter y independent of variable x, as evaluated at x=1.

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A330498 Expansion of e.g.f. Sum_{k>=1} log(1 + log(1/(1 - x))^k) / k.

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A373856 a(n) = Sum_{k=1..n} k! * k^(2n-1) |Stirling1(n,k)|.

A277408 Triangle, read by rows, where the g.f. of row n equals the sum of permutations of compositions of functions (1 + kyx) for k=1..n with parameter y independent of variable x, as evaluated at x=1.

A341589 a(n) = Sum_{k=n..2n} |Stirling1(2n, k) * Stirling1(k, n)|.