A052517 -id:A052517 - cp's OEIS frontend

A028421 Triangle read by rows: T(n, k) = (k+1)*A132393(n+1, k+1), for 0 <= k <= n.

1, 1, 2, 2, 6, 3, 6, 22, 18, 4, 24, 100, 105, 40, 5, 120, 548, 675, 340, 75, 6, 720, 3528, 4872, 2940, 875, 126, 7, 5040, 26136, 39396, 27076, 9800, 1932, 196, 8, 40320, 219168, 354372, 269136, 112245, 27216, 3822, 288, 9

Offset: 0

Peter Wiggen (wiggen(AT)math.psu.edu)

Previous name was: Number triangle f(n, k) from n-th differences of the sequence {1/m^2}{m >= 1}, for n >= 0; the n-th difference sequence is {(-1)^n*n!*P(n, m)/D(n, m)^2}{m >= 1} where P(n, x) is the row polynomial P(n, x) = Sum_{k=0..n} f(n,k)*x^k and D(n, x) = x*(x+1)*...*(x+n).
From Johannes W. Meijer, Oct 07 2009: (Start)
The higher-order exponential integrals E(x,m,n) are defined in A163931 and the general formula of the asymptotic expansion of E(x,m,n) can be found in A163932.
We used the general formula and the asymptotic expansion of E(x,m=1,n), see A130534, to determine that E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2 + 6*n + 3*n^2)/x^2 - (6 + 22*n + 18*n^2 + 4*n^3)/x^3 + ...) which can be verified with the EA(x,2,n) formula, see A163932. The coefficients in the denominators of this expansion lead to the sequence given above.
The asymptotic expansion of E(x,m=2,n) leads for n from one to ten to known sequences, see the cross-references. With these sequences one can form the triangles A165674 (left hand columns) and A093905 (right hand columns).
(End)
For connections to an operator relation between log(x) and x^n(d/dx)^n, see A238363. - Tom Copeland, Feb 28 2014
From Wolfdieter Lang, Nov 25 2018: (Start)
The signed triangle t(n, k) := (-1)^{n-k}*f(n, k) gives (n+1)*N(-1;n,x) = Sum_{k=0..n} t(n, k)*x^k, where N(-1;n,x) are the Narumi polynomials with parameter a = -1 (see the Weisstein link).
The members of the n-th difference sequence of the sequence {1/m^2}_{m>=1} mentioned above satisfies the recurrence delta(n, m) = delta(n-1, m+1) - delta(n-1, m), for n >= 1, m >= 1, with input delta(0, m) = 1/m^2. The solution is delta(n, m) = (n+1)!*N(-1;n,-m)/risefac(m, n+1)^2, with Narumi polynomials N(-1;n,x) and the rising factorials risefac(x, n+1) = D(n, x) = x*(x+1)*...*(x+n).
The above mentioned row polynomials P satisfy P(n, x) = (-1)^n*(n + 1)*N(-1;n,-x), for n >= 0. The recurrence is P(n, x) = (-x^2*P(n-1, x+1) + (n+x)^2*P(n-1, x))/n, for n >= 1, and P(0, x) = 1. (End)
The triangle is the exponential Riordan square (cf. A321620) of -log(1-x) with an additional main diagonal of zeros. - Peter Luschny, Jan 03 2019

			The triangle T(n, k) begins:
n\k       0        1        2        3        4       5       6      7     8   9 10
------------------------------------------------------------------------------------
0:        1
1:        1        2
2:        2        6        3
3:        6       22       18        4
4:       24      100      105       40        5
5:      120      548      675      340       75       6
6:      720     3528     4872     2940      875     126       7
7:     5040    26136    39396    27076     9800    1932     196      8
8:    40320   219168   354372   269136   112245   27216    3822    288     9
9:   362880  2053152  3518100  2894720  1346625  379638   66150   6960   405  10
10: 3628800 21257280 38260728 33638000 17084650 5412330 1104411 145200 11880 550 11
... - _Wolfdieter Lang_, Nov 23 2018

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric Weisstein's World of Mathematics, Narumi Polynomial [here for a = -1].

Row sums give A000254(n+1), n >= 0.
Cf. A132393 (unsigned Stirling1), A061356, A139526, A321620.
From Johannes W. Meijer, Oct 07 2009: (Start)
A000142, A052517, 3*A000399, 5*A000482 are the first four left hand columns; A000027, A002411 are the first two right hand columns.
The asymptotic expansion of E(x,m=2,n) leads to A000254 (n=1), A001705 (n=2), A001711 (n=3), A001716 (n=4), A001721 (n=5), A051524 (n=6), A051545 (n=7), A051560 (n=8), A051562 (n=9), A051564 (n=10), A093905 (triangle) and A165674 (triangle).
Cf. A163931 (E(x,m,n)), A130534 (m=1), A163932 (m=3), A163934 (m=4), A074246 (E(x,m=2,n+1)). (End)

A028421 := proc(n,k) (-1)^(n+k)*(k+1)*Stirling1(n+1,k+1) end:
seq(seq(A028421(n,k), k=0..n), n=0..8);
# Johannes W. Meijer, Oct 07 2009, Revised Sep 09 2012
egf := (1 - t)^(-x - 1)*(1 - x*log(1 - t)):
ser := series(egf, t, 16): coefft := n -> expand(coeff(ser,t,n)):
seq(seq(n!*coeff(coefft(n), x, k), k = 0..n), n = 0..8); # Peter Luschny, Jun 12 2022

f[n_, k_] = (k + 1) StirlingS1[n + 1, k + 1] // Abs; Flatten[Table[f[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-François Alcover, Jun 01 2011, after formula *)

# uses[riordan_square from A321620]
riordan_square(-ln(1 - x), 10, True) # Peter Luschny, Jan 03 2019

E.g.f.: d/dt(-log(1-t)/(1-t)^x). - Vladeta Jovovic, Oct 12 2003
The e.g.f. with offset 1: y = x + (1 + 2*t)*x^2/2! + (2 + 6*t + 3*t^2)*x^3/3! + ... has series reversion with respect to x equal to y - (1 + 2*t)*y^2/2! + (1 + 3*t)^2*y^3/3! - (1 + 4*t)^3*y^4/4! + .... This is an e.g.f. for a signed version of A139526. - Peter Bala, Jul 18 2013
Recurrence: T(n, k) = 0 if n < k; if k = 0 then T(0, 0) = 1 and T(n, 0) = n * T(n-1, 0) for n >= 1, otherwise T(n, k) = n*T(n-1, k) + ((k+1)/k)*T(n-1, k-1). From the unsigned Stirling1 recurrence. - Wolfdieter Lang, Nov 25 2018

Edited by Wolfdieter Lang, Nov 23 2018

A225479 Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 22, 36, 24, 0, 24, 100, 210, 240, 120, 0, 120, 548, 1350, 2040, 1800, 720, 0, 720, 3528, 9744, 17640, 21000, 15120, 5040, 0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320, 0, 40320, 219168, 708744, 1614816

Offset: 0

Peter Luschny, May 20 2013

The Digital Library of Mathematical Functions defines the Stirling cycle numbers as (-1)^(n-k) times the Stirling numbers of the first kind.

			[n\k][0,   1,   2,    3,    4,    5,   6]
[0]   1,
[1]   0,   1,
[2]   0,   1,   2,
[3]   0,   2,   6,    6,
[4]   0,   6,  22,   36,   24,
[5]   0,  24, 100,  210,  240,  120,
[6]   0, 120, 548, 1350, 2040, 1800, 720.
...
T(4,2) = 22: The table below shows the compositions of 4 into two parts.
n = 4    Composition       Weight     4!*Weight
            3 + 1            1/3         8
            1 + 3            1/3         8
            2 + 2          1/2*1/2       6
                                        = =
                                  total 22

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 245.

Vincenzo Librandi, Rows n = 0..50, flattened
Digital Library of Mathematical Functions, Set Partitions: Stirling Numbers
S. Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients J. Integer. Seq., Vol. 16 (2013), Article 13.1.3

Cf. A048594 (signed version without the first column), A132393.

Columns k=0..6 are A000007, A104150, A052517, A052748, A052753, A052767, A052779.

A225479 := proc(n, k) option remember;
if k > n or  k < 0 then return(0) fi;
if n = 0 and k = 0 then return(1) fi;
k*A225479(n-1, k-1) + (n-1)*A225479(n-1, k) end;
for n from 0 to 9 do seq(A225479(n, k), k = 0..n) od;

t[n_, k_] := k!*StirlingS1[n, k] // Abs; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2013 *)

T(n,k)={k!*abs(stirling(n,k,1))} \\ Andrew Howroyd, Jul 27 2020

def A225479(n, k): return factorial(k)*stirling_number1(n, k)
for n in (0..6): [A225479(n,k) for k in (0..n)]

For a recursion see the Maple program.
T(n, 0) = A000007; T(n, 1) = A000142; T(n, 2) = A052517.
T(n, 3) = A052748; T(n, n) = A000142; T(n, n-1) = A001286.
row sums = A007840; alternating row sums = A006252.
From Peter Bala, Sep 20 2013: (Start)
E.g.f.: 1/(1 + x*log(1 - t)) = 1 + x*t + (x + 2*x^2)*t^2/2! + (2*x + 6*x^2 + 6*x^3)*t^3/3! + ....
T(n,k) = n!*( the sum of the total weight of the compositions of n into k parts where each part i has weight 1/i ) (see Eger, Theorem 1). An example is given below.  (End)
T(n,k) = A132393(n,k) * A000142(k). - Philippe Deléham, Jun 24 2015

A302827 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.

Original entry on oeis.org

0, 0, 4, 18, 164, 2600, 64072, 2272032, 109735488, 6930012672, 554528623104, 54840436992000, 6568892183808000, 937223951339520000, 157057344897601536000, 30545188599606047539200, 6823697557721234964480000, 1735362552287102663393280000

Offset: 0

Vaclav Kotesovec, May 15 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A052517, A304581, A304582, A304589, A304654, A370225, A370226.

List([0..20],n->Factorial(n)^2*Sum([1..n-1],k->1/(k*(n-k))^2)); # Muniru A Asiru, May 16 2018

seq(factorial(n)^2*add(1/(k*(n-k))^2,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

Table[n!^2*Sum[1/(k*(n-k))^2, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[2, x]^2, {x, 0, 20}], x] * Range[0,20]!^2

Recurrence: n*(2*n - 3)*a(n) = (n-1)*(6*n^3 - 25*n^2 + 33*n - 12)*a(n-1) - (n-2)^3*(6*n^3 - 29*n^2 + 42*n - 15)*a(n-2) + (n-3)^4*(n-2)^3*(2*n - 1)*a(n-3).

a(n) / (n!)^2 ~ Pi^2/(3*n^2) + 4*log(n)/n^3.

A052748 Expansion of e.g.f.: -(log(1-x))^3.

Original entry on oeis.org

0, 0, 0, 6, 36, 210, 1350, 9744, 78792, 708744, 7036200, 76521456, 905507856, 11589357312, 159580302336, 2352940786944, 36994905688320, 617953469022720, 10929614667747840, 204073497562936320, 4011658382046919680, 82822558521844224000, 1791791417179298304000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Original name: A simple grammar.

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 704

Column k=3 of A225479.

Cf. A000399, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combinat):seq(stirling1(j, 3)*3!*(-1)^(j+1), j=0..50); # Leonid Bedratyuk, Aug 07 2012

a(n) = {3!*stirling(n,3,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(1/(1-x))^3.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (-1 - 3*n - 3*n^2 - n^3)*a(n+1) + (9*n + 7 + 3*n^2)*a(n+2) + (-6 - 3*n)*a(n+3) + a(n+4)}.
a(n) = stirling1(n, 3)*3!*(-1)^(n+1). - Leonid Bedratyuk, Aug 07 2012
a(n) = 6*A000399(n). - Andrew Howroyd, Jul 27 2020

Name changed and terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A101613 (2n)! * Sum[k=n..2n, 1/k].

Original entry on oeis.org

3, 26, 684, 35664, 3068640, 392722560, 69878833920, 16484477184000, 4976250951168000, 1870345490614272000, 856314330237628416000, 469109144339151224832000, 302956190548293037916160000

Offset: 1

Ralf Stephan, Dec 10 2004

Cf. A101609, A101610, A101611, A101612, A000254, A046673, A052517.

A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

Original entry on oeis.org

0, 0, 8, 54, 1240, 70000, 7941968, 1589632128, 512918521344, 249820864339968, 174720109813751808, 168721560082538496000, 217977447876560510976000, 367117517435096337481728000, 788739873984137255456342016000, 2122296978948474538763602624512000

Offset: 0

Vaclav Kotesovec, May 15 2018

nonn

In general, for m > 1, Sum_{k=1..n-1} 1/(k*(n-k))^m is asymptotic to 2*Zeta(m)/n^m.

Seiichi Manyama, Table of n, a(n) for n = 0..150
Eric Weisstein's World of Mathematics, Polylogarithm.

Cf. A052517, A302827, A002547, A002548, A304581, A304582, A304655.

List([0..20],n->Factorial(n)^3*Sum([1..n-1],k->1/(k*(n-k))^3)); # Muniru A Asiru, May 16 2018

seq(factorial(n)^3*add(1/(k*(n-k))^3,k=1..n-1),n=0..20); # Muniru A Asiru, May 16 2018

Table[n!^3*Sum[1/(k*(n-k))^3, {k, 1, n-1}], {n, 0, 20}]
CoefficientList[Series[PolyLog[3, x]^2, {x, 0, 20}], x] * Range[0,20]!^3

Recurrence: n^2*(12*n^4 - 108*n^3 + 354*n^2 - 501*n + 260)*a(n) = 2*(n-1)^2*(24*n^7 - 306*n^6 + 1620*n^5 - 4599*n^4 + 7516*n^3 - 7015*n^2 + 3444*n - 696)*a(n-1) - 6*(n-2)^5*(12*n^7 - 162*n^6 + 906*n^5 - 2700*n^4 + 4583*n^3 - 4378*n^2 + 2163*n - 436)*a(n-2) + 2*(n-3)^5*(n-2)^3*(24*n^7 - 342*n^6 + 2004*n^5 - 6201*n^4 + 10816*n^3 - 10497*n^2 + 5208*n - 1048)*a(n-3) - (n-4)^6*(n-3)^5*(n-2)^3*(12*n^4 - 60*n^3 + 102*n^2 - 69*n + 17)*a(n-4).

a(n) / (n!)^3 ~ 2*Zeta(3)/n^3.

A052753 Expansion of e.g.f.: log(1-x)^4.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824, 33936357312, 487530074304, 7463742249600, 121367896891776, 2089865973021696, 37999535417459712, 727710096185266176, 14642785817771802624, 308902349883623731200, 6818239581643475251200

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..448
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 709

Column k=4 of A225479.

Cf. A000454, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[(Log[1-x])^4, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

x='x+O('x^30); concat(vector(4), Vec(serlaplace((log(1-x))^4))) \\ G. C. Greubel, Aug 30 2018

a(n) = {4!*stirling(n,4,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^4.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, (1+4*n+6*n^2+4*n^3+n^4)*a(n+1) + (-4*n^3-15-18*n^2-28*n)*a(n+2) + (6*n^2+24*n+25)*a(n+3) + (-4*n-10)*a(n+4)+a(n+5), a(3)=0, a(4)=24}.
a(n) ~ (n-1)! * 2*log(n)*(2*log(n)^2 + 6*gamma*log(n) - Pi^2 + 6*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013
a(n) = 24*A000454(n) = 4!*(-1)^n*Stirling1(n,4). - Andrew Howroyd, Jul 27 2020

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A101609 a(n) = n! * Sum_{k=1..floor(n/2)} 1/k.

Original entry on oeis.org

0, 2, 6, 36, 180, 1320, 9240, 84000, 756000, 8285760, 91143360, 1173553920, 15256200960, 226040855040, 3390612825600, 56865153945600, 966707617075200, 18112111963545600, 344130127307366400, 7125892746964992000

Offset: 1

Ralf Stephan, Dec 10 2004

Cf. A101610, A101611, A101612, A101613, A000254, A046673, A052517.

Table[ n! HarmonicNumber[ Floor[ n/2]], {n, 20}] (* Robert G. Wilson v, Dec 11 2004 *)

a(1) = 0, a(n) = a(n-1)*n + (1 + (-1)^n)*(n-1)!. - Daniel Suteu, Feb 06 2017

A101610 n! * Sum[k=1..ceiling(n/2), 1/k].

Original entry on oeis.org

1, 2, 9, 36, 220, 1320, 10500, 84000, 828576, 8285760, 97796160, 1173553920, 16145775360, 226040855040, 3554072121600, 56865153945600, 1006228442419200, 18112111963545600, 356294637348249600

Offset: 1

Ralf Stephan, Dec 10 2004

Cf. A101609, A101611, A101612, A101613, A000254, A046673, A052517.

Table[n!Sum[1/k,{k,1,Ceiling[n/2]}],{n,20}]  (* Harvey P. Dale, Apr 25 2011 *)

A101611 a(n) = n! * Sum_{k=ceiling(n/2)..n} 1/k.

Original entry on oeis.org

1, 3, 5, 26, 94, 684, 3828, 35664, 270576, 3068640, 29400480, 392722560, 4546558080, 69878833920, 948550176000, 16484477184000, 256697973504000, 4976250951168000, 87435019510272000, 1870345490614272000

Offset: 1

Ralf Stephan, Dec 10 2004

Cf. A101609, A101610, A101612, A101613, A000254, A046673, A052517.

Rest@Table[CoefficientList[Series[(Log[1-x]-x*Log[1-x^2])/(x-1),{x, 0, 20}],x][[n]](n-1)!,{n, 1, 20}] (* Benedict W. J. Irwin, Apr 25 2017 *)

a(n) = n! * sum(k=ceil(n/2), n, 1/k); \\ Michel Marcus, Apr 25 2017

import math
from sympy import factorial
def a(n): return factorial(n)*sum([1/k for k in range(int(math.ceil(n/2)), n + 1)]) # Indranil Ghosh, Apr 25 2017

E.g.f: (log(1 - x) - x*log(1 - x^2))/(x - 1). - Benedict W. J. Irwin, Apr 25 2017

cp's OEIS Frontend

A028421 Triangle read by rows: T(n, k) = (k+1)*A132393(n+1, k+1), for 0 <= k <= n.

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A225479 Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

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A302827 a(n) = (n!)^2 * Sum_{k=1..n-1} 1/(k*(n-k))^2.

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A052748 Expansion of e.g.f.: -(log(1-x))^3.

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A101613 (2n)! * Sum[k=n..2n, 1/k].

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A304589 a(n) = (n!)^3 * Sum_{k=1..n-1} 1/(k*(n-k))^3.

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A052753 Expansion of e.g.f.: log(1-x)^4.

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