A000399 -id:A000399 - cp's OEIS frontend

A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

1, -2, 25, -75, 1099, -4340, 79064, -382060, 8550916, -48306984, 1303568760, -8346754416, 266955481584, -1894529909376, 70785236377728, -547468189825536, 23610353987137536, -196402650598402560, 9679304091074250240, -85687212859582878720, 4785340778000524477440

Offset: 3

Seiichi Manyama, Sep 26 2021

sign

Column k=3 of A190782.

Cf. A000399, A000454, A054651, A081051, A028340, A347987, A348063, A348065, A348068.

a(n) = n!*polcoef(sum(k=3, n, binomial(x, k)), 3);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^3/(6*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348064(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**3) for k in range(3,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^3/(6 * (1 - x)).

A081051 Stirling numbers of the first kind.

Original entry on oeis.org

0, 0, 1, -6, 35, -225, 1624, -13132, 118124, -1172700, 12753576, -150917976, 1931559552, -26596717056, 392156797824, -6165817614720, 102992244837120, -1821602444624640, 34012249593822720, -668609730341153280, 13803759753640704000, -298631902863216384000

Offset: 0

Paul Barry, Mar 05 2003

Coefficient of x^3 in Product {k=0..(n-1), x-k}.

Cf. A000399, A008275, A081048.

Table[StirlingS1[n, 3], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2022 *)

E.g.f. (1+x)^(-1)*log(1+x)^2/2
a(n) = (-1)^n*det(S(i+3,j+2), 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind and n>1. [Mircea Merca, Apr 06 2013]
a(n) ~ n! * (-1)^n * log(n)^2/2 * (1 + 2*gamma/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 03 2022

A081052 Difference of Stirling numbers of the first kind.

Original entry on oeis.org

0, 1, -4, 17, -85, 499, -3388, 26200, -227708, 2199276, -23382216, 271461816, -3418002432, 46399476096, -675622445184, 10504980616320, -173726527230720, 3045008035203840, -56389237652344320, 1100174877158791680, -22556707790402304000, 484876713643386624000

Offset: 0

Paul Barry, Mar 05 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000399, A081051, A081048.

Table[StirlingS1[n,2]-StirlingS1[n,3],{n,30}] (* Harvey P. Dale, May 02 2012 *)

for(n=1, 22, print1(stirling(n, 2) - stirling(n, 3),", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = s(n,2) - s(n,3), s(n,m) = signed Stirling number of the first kind.
E.g.f. (1+x)^-1 * (log(1+x) - (log(1+x)^2)/2).
Conjecture: a(n) +3*(n-1)*a(n-1) +(3*n^2-9*n+7)*a(n-2) +(n-2)^3*a(n-3)=0. - R. J. Mathar, Nov 24 2012

A309237 Records in the indices of largest unsigned Stirling number of first kind: a(n) = smallest m such that c(m,n) = max_{k=0,1...,m} c(m,k).

Original entry on oeis.org

0, 1, 2, 8, 25, 73, 204, 565, 1557, 4275, 11710, 32022, 87464, 238692, 650971, 1774466

Offset: 0

Max Alekseyev, Jul 17 2019

Smallest m such that A065048(m-1) = c(m,n).
For k in the interval [a(n),a(n+1)-1], A065048(k-1) = c(k,n).
Ratio a(n+1)/a(n) seems to decrease and tend to exp(1) as n grows.

			n=2 is a value for index k delivering the maximum value of c(m,k) for each fixed m in the interval [a(2),a(3)-1] = [2,7]. Then, for m in [a(3),a(4)-1] = [8,24], the maximum is given by c(m,3), and so on.

Cf. A000254, A000399, A008275, A065048.

{ A309237(n) = my(t=prod(i=1,n-1,x+i+O(x^n)), m=n); while( polcoef(t,n-1)-polcoef(t,n-2) < 0, t*=x+m; m++); m; }

a(14)-a(15) from Alexander Fritsch and Johann Peters, Dec 04 2024

A346945 Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

Original entry on oeis.org

1, -6, 35, -235, 1834, -16352, 164044, -1830630, 22513326, -302700926, 4419167532, -69637654996, 1178377833424, -21315571470320, 410529985172400, -8388475139138320, 181270810764205440, -4130796696683135280, 99008773205008777760, -2490134250475836315120

Offset: 3

Ilya Gutkovskiy, Aug 08 2021

sign

Cf. A000399, A003713, A081051, A346944, A346946, A346947.

Cf. A346975.

nmax = 22; CoefficientList[Series[Log[1 + Log[1 + x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS1[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 22}]

a(n) = Stirling1(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,3) * k * a(k).

a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling1(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A383164 Expansion of e.g.f. -log(1 - (exp(2*x) - 1)/2)^3 / 6.

Original entry on oeis.org

0, 0, 0, 1, 18, 255, 3555, 52290, 831684, 14405580, 271688580, 5562400800, 123123764808, 2933953637472, 74953425290016, 2044855241694720, 59361121229581440, 1827578437315965696, 59494057195888597248, 2042194772007257103360, 73731225467600254686720

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Column k=3 of A383149.

Cf. A000399, A383166.

a(n) = sum(k=3, n, 2^(n-k)*stirling(n, k, 2)*abs(stirling(k, 3, 1)));

a(n) = Sum{k=3..n} 2^(n-k) * Stirling2(n,k) * |Stirling1(k,3)|.

a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n - 1/2) * log(n)^2 / (exp(n) * log(3)^n). - Vaclav Kotesovec, Apr 18 2025

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

A381106 Expansion of e.g.f. -log(1-x)^3 * (exp(x) - 1) / 6.

Original entry on oeis.org

0, 0, 0, 0, 4, 40, 320, 2555, 21728, 200802, 2024510, 22221485, 264453750, 3396686865, 46873789235, 692049842575, 10889098371032, 181952854080860, 3218431205690356, 60087159752141449, 1180916015576750386, 24372799835934758327, 527084149497398472485, 11919591185373007970251

Offset: 0

Seiichi Manyama, Feb 14 2025

nonn

Cf. A052863, A381105.

Cf. A000399, A381022.

a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k, 3, 1)));

a(n) = Sum_{k=0..n-1} binomial(n,k) * |Stirling1(k,3)|.

a(n) = A381022(n) - A000399(n).

A381108 Expansion of e.g.f. log(1-x)^2 * (exp(x) - 1) / (2 * (1-x)).

Original entry on oeis.org

0, 0, 0, 3, 30, 245, 2010, 17549, 165942, 1705584, 19024275, 229478689, 2981315139, 41545542818, 618579336284, 9804891730633, 164897938095108, 2933486106772376, 55047126101826453, 1086816606230786217, 22523274090016854661, 488907589907823010158, 11093875133012393113766

Offset: 0

Seiichi Manyama, Feb 14 2025

nonn

Cf. A002627, A381107.

Cf. A000399, A381024.

a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k+1, 3, 1)));

a(n) = Sum_{k=0..n-1} binomial(n,k) * |Stirling1(k+1,3)|.

a(n) = A381024(n) - A000399(n+1).

A081103 Alternating sum of first three Stirling numbers of the first kind.

Original entry on oeis.org

0, 1, -4, 18, -95, 584, -4123, 32969, -294992, 2922956, -31791716, 376719892, -4832017320, 66713229192, -986611705584, 15561976320144, -260804276106624, 4628322010931328, -86710491660063744, 1710290952899283456, -35427639035553292800, 768970029545198092800

Offset: 0

Paul Barry, Mar 05 2003

Cf. A000399, A081052, A081046.

a(n) = s(n, 1)-s(n, 2)+s(n, 3), s(n, m)= signed Stirling numbers of the first kind.

E.g.f.: (1+x)^(-1)*(log(1+x)-log(1+x)^2/2+log(1+x)^3/6).

cp's OEIS Frontend

A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Python

Formula

A081051 Stirling numbers of the first kind.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A081052 Difference of Stirling numbers of the first kind.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A309237 Records in the indices of largest unsigned Stirling number of first kind: a(n) = smallest m such that c(m,n) = max_{k=0,1...,m} c(m,k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A346945 Expansion of e.g.f. log( 1 + log(1 + x)^3 / 3! ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A383164 Expansion of e.g.f. -log(1 - (exp(2*x) - 1)/2)^3 / 6.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

Formula

Extensions

A381106 Expansion of e.g.f. -log(1-x)^3 * (exp(x) - 1) / 6.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula