A130534 -id:A130534 - cp's OEIS frontend

A001725 a(n) = n!/5!.

1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840, 726485760, 10897286400, 174356582400, 2964061900800, 53353114214400, 1013709170073600, 20274183401472000, 425757851430912000, 9366672731480064000, 215433472824041472000, 5170403347776995328000

Offset: 5

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=6) ~ exp(-x)/x*(1 - 6/x + 42/x^2 - 336/x^3 + 3024/x^4 - 30240/x^5 + 332640/x^6 - 3991680/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 5..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 265.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

a(n)= A049374(n-4), n >= 1 (first column of triangle). Cf. A049460, A051339. a(n)= A051338(n-5, 0)*(-1)^(n-1) (first unsigned column of triangle).

Cf. A245334, A000142.

a001725 = (flip div 120) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/120: n in [5..25]]; // Vincenzo Librandi, Jul 20 2011

lst={};Do[AppendTo[lst, n!/5! ], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
Range[5,30]!/120 (* Harvey P. Dale, Dec 20 2014 *)

a(n)=n!/120 \\ Charles R Greathouse IV, Jul 19 2011

E.g.f. if offset 0: 1/(1-x)^6.
a(n) = A173333(n,5). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+6)/(x*(k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0)/(40*x^2) -1/(20*x^2) -1/(5*x) , where W(k) = 1 + 1/( 1 - x*(k+4)/( x*(k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n,n-5) / 6. - Reinhard Zumkeller, Aug 31 2014
E.g.f.: x^5 / (5! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=5} 1/a(n) = 120*e - 325.
Sum_{n>=5} (-1)^(n+1)/a(n) = 45 - 120/e. (End)

More terms from Harvey P. Dale, Dec 20 2014

A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

Original entry on oeis.org

0, 0, 0, 0, 24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024

Offset: 0

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

Also, starting with n=4, the square of area of cyclic quadrilateral with sides n, n-1, n-2, n-3. - Zak Seidov, Jun 20 2003
Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - Eric M. Schmidt, Oct 31 2012
Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
Number of 4-permutations of the set {1, 2, ..., n}. - Joerg Arndt, Apr 05 2014

Eric M. Schmidt, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 719
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Wikipedia, Pochhammer symbol.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001094, A002378, A007531, A052787.

[n*(n-1)*(n-2)*(n-3): n in [0..30]]; // G. C. Greubel, Nov 19 2017

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,4), n=0..34); # Zerinvary Lajos, Apr 26 2007
G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 34 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..34); # Zerinvary Lajos, Apr 05 2009

Table[n*(n+1)*(n+2)*(n+3), {n,-3,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Times@@@Partition[Range[-3,60], 4, 1] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[ {5,-10,10,-5,1}, {0,0,0,0,24}, 60] (* Harvey P. Dale, May 09 2012 *)

A052762(n):=n*(n-1)*(n-2)*(n-3)$
makelist(A052762(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=24*binomial(n,4) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).
a(n) = A001094(n) - n.
E.g.f.: x^4*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.
a(n) + 1 = A062938(n-4) for n > 4. - Amarnath Murthy, Dec 13 2003
a(n) = numbperm(n, 4). - Zerinvary Lajos, Apr 26 2007
O.g.f.: -24*x^4/(-1+x)^5. - R. J. Mathar, Nov 23 2007
For n > 4: a(n) = A173333(n, n-4). - Reinhard Zumkeller, Feb 19 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24. - Harvey P. Dale, May 09 2012
a(n) = a(n-1) + 4*A007531(n). - J. M. Bergot, May 30 2012
a(n) = (n)4 = Pochhammer(n,4), using the "falling factorial" convention; other authors write Pochhammer(x,k) for what is denoted x^(k) in the Wikipedia article, then a(n) = (n-3)^(4). - _M. F. Hasler, Oct 20 2013
a(n) - 1 = A069756(n-2) for n >= 4. - Jean-Christophe Hervé, Nov 01 2015
a(n) = 24 * A000332(n). - Bruce J. Nicholson, Apr 03 2017
From R. J. Mathar, Jun 30 2021: (Start)
Sum_{n>=4} 24*(-1)^n/a(n) = A242023.
Sum_{n>=4} 1/a(n) = 1/18. (End)

More terms from Henry Bottomley, Mar 20 2000

Formula corrected by Philippe Deléham, Dec 12 2003

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

Original entry on oeis.org

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

A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

Original entry on oeis.org

1, 3, 3, 11, 18, 6, 50, 105, 60, 10, 274, 675, 510, 150, 15, 1764, 4872, 4410, 1750, 315, 21, 13068, 39396, 40614, 19600, 4830, 588, 28, 109584, 354372, 403704, 224490, 68040, 11466, 1008, 36, 1026576, 3518100, 4342080, 2693250, 949095, 198450

Offset: 1

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Oct 22 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The general formula for the asymptotic expansion E(x,m,n) ~ E(x,m-1,n+1)/x - n*E(x,m-1,n+2)/x^2 + n*(n+1) * E(x,m-1,n+3)/x^3 - n*(n+1)*(n+2)*E(x,m-1,n+4)/x^4 + ...., m >= 1 and n >= 1.
We used this formula and the asymptotic expansion of E(x,m=2,n), see A028421, to determine that E (x,m=3,n) ~ (exp(-x)/x^3)*(1 - (3+3*n)/x + (11+18*n+6*n^2)/x^2 - (50+105*n+ 60*n^2+ 10*n^3)/x^3 + .. ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to ten to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A001879, see A163938 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=3,n).

			The first few rows of the triangle are:
[1]
[3, 3]
[11, 18, 6]
[50, 105, 60, 10]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A163931 (E(x,m,n)) and A163938.
Cf. A048994 (Stirling1), A000399 (row sums).
A000254, 3*A000399, 6*A000454, 10*A000482, 15*A001233, 21*A001234 equal the first six left hand columns.
A000217, A006011 and A163933 equal the first three right hand columns.
The asymptotic expansion leads to A000399 (n=1), A001706 (n=2), A001712 (n=3), A001717 (n=4), A001722 (n=5), A051525 (n=6), A051546 (n=7), A051561 (n=8), A051563 (n=9) and A051565 (n=10).
Cf. A130534 (m=1), A028421 (m=2) and A163934 (m=4).

nmax:=8; with(combinat): for n1 from 1 to nmax do for m from 1 to n1 do a(n1, m) := (-1)^(n1+m)*binomial(m+1, 2)*stirling1(n1+1, m+1) od: od: seq(seq(a(n1,m), m=1..n1), n1=1..nmax);
# End program 1
with(combinat): imax:=6; EA:=proc(x, m, n) local E, i; E := 0: for i from m-1 to imax+1 do E := E + sum((-1)^(m+k1+1)*binomial(k1, m-1)*n^(k1-m+1)* stirling1(i, k1), k1=m-1..i)/x^(i-m+1) od: E := exp(-x)/x^(m)*E: return(E); end: EA(x, 3, n);
# End program 2

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+1, 2] * StirlingS1[n+1, m+1]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 42]] (* Jean-François Alcover, Jun 01 2011, after formula *)

for(n=1,10, for(m=1,n, print1((-1)^(n+m)*binomial(m+1,2) *stirling(n+1,m+1,1), ", "))) \\ G. C. Greubel, Aug 08 2017

a(n,m) = (-1)^(n+m)*binomial(m+1,2)*stirling1(n+1,m+1) for n >= 1 and 1 <= m <= n.

Edited by Johannes W. Meijer, Sep 22 2012

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.

Original entry on oeis.org

0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, 278256, 314160, 353430, 396270, 442890, 493506, 548340, 607620

Offset: 0

N. J. A. Sloane

Non-vanishing diagonal of (A132440)^4/4. Third subdiagonal of unsigned A238363 without the zero. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of the complete graph K_4. - Tom Copeland, Apr 05 2014
Total number of pips on a set of trominoes (3-armed dominoes) with up to n pips on each arm. - Alan Shore and N. J. A. Sloane, Jan 06 2016
Also the number of minimum connected dominating sets in the (n+2)-crown graph. - Eric W. Weisstein, Jun 29 2017
Crossing number of the (n+3)-cocktail party graph (conjectured). - Eric W. Weisstein, Apr 29 2019
Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - Edward Krogius, Jul 31 2022

			G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Vincenzo Librandi, Table of n, a(n) for n = 0..690
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Aleksandar Petojević and Nenad Đapić, The vAm(a,b,c;z) function, Preprint 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Hasan Unal, Proof Without Words: Sums of Products of Three Consecutive Integers, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Crown Party Graph.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Index entries for sequences related to Bessel functions or polynomials
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000332, A002378, A033486, A033488, A034827, A050534, A130534, A132440, A238363.
Partial sums of A007531.
A row of the array in A129533.
A column of the triangle in A331430.
Sequences of the form binomial(n+k,k)*binomial(n+k+2,k): A000012 (k=0), A005563 (k=1), this sequence (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).

[n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(binomial(n+3,4)*6, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* Harvey P. Dale, Nov 27 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* Harvey P. Dale, Nov 27 2013 *)
Table[6 Binomial[n+3, 4], {n,0,40}] (* Eric W. Weisstein, Jun 29 2017 *)
Times @@@ Table[n+k, {n, 0, 40}, {k, 0, 3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n)=6*binomial(n+3,4) \\ Charles R Greathouse IV, Apr 17 2012

concat(0, Vec(6*x/(1-x)^5 + O(x^100))) \\ Altug Alkan, Nov 29 2015

def A033487(n): return 6*binomial(n+3,4)
print([A033487(n) for n in range(41)]) # G. C. Greubel, Feb 08 2025

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)
G.f.: 6*x/(1-x)^5.
a(n) = 6*binomial(n+3, 4) = 6*A000332(n+3).
a(n) = a(n-1) + A007531(n+1).
a(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)
Constant term in Bessel polynomial {y_n(x)}''.
a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - Zerinvary Lajos, May 25 2005
a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - Zerinvary Lajos, May 17 2006
From Zerinvary Lajos, May 11 2007: (Start)
a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j.
a(n) = Sum_{j=1..n} j*(n+2)*(n-1)/2. (End)
Sum_{n>0} 1/a(n) = 2/9. - Enrique Pérez Herrero, Nov 10 2013
a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - Michael Somos, Apr 06 2014
a(n) = A002378(binomial(n+2,2)-1). - Salvador Cerdá, Nov 04 2016
a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - Michel Marcus, Oct 29 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/4. - Stefano Spezia, Jul 03 2025

A001730 a(n) = n!/6!.

Original entry on oeis.org

1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960, 1816214400, 29059430400, 494010316800, 8892185702400, 168951528345600, 3379030566912000, 70959641905152000, 1561112121913344000, 35905578804006912000, 861733891296165888000, 21543347282404147200000

Offset: 6

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=7) ~ exp(-x)/x*(1 - 7/x + 56/x^2 - 504/x^3 + 5040/x^4 - 55440/x^5 + 665280/x^6 - 8648640/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 6..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 266.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling. II, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 107-108 1963 1-77.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A051338, A051339, A051379.

Cf. A130534, A163931, A173333, A245334, A000142.

a001730 = (flip div 720) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/720: n in [6..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_]:=n!/6!; Array[a,4!,6] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2009 *)

a(n)=n!/720 \\ Charles R Greathouse IV, Jan 12 2012

a(n)= A051339(n-6, 0)*(-1)^n (first unsigned column of triangle).
E.g.f.: x^6/(6!*(1-x)). [corrected by Alois P. Heinz, Jul 09 2021]
a(n) = A173333(n,6). - Reinhard Zumkeller, Feb 19 2010
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(k+7)/(x*(k+7) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
a(n) = A245334(n,n-6) / 7. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=6} 1/a(n) = 720*e - 1956.
Sum_{n>=6} (-1)^n/a(n) = 720/e - 264. (End)

A049388 a(n) = (n+7)!/7!.

Original entry on oeis.org

1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 4151347200, 70572902400, 1270312243200, 24135932620800, 482718652416000, 10137091700736000, 223016017416192000, 5129368400572416000, 123104841613737984000, 3077621040343449600000, 80018147048929689600000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=8) ~ exp(-x)/x*(1 - 8/x + 72/x^2 - 720/x^3 + 7920/x^4 - 95040/x^5 + 235520/x^6 - 17297280/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

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

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A051339, A051379.

Cf. A130534, A163931, A173333, A245334.

a049388 = (flip div 5040) . a000142 . (+ 7)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+7)/5040: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

((Range[0,20]+7)!)/7! (* Harvey P. Dale, Jul 31 2012 *)

vector(20,n,n--; (n+7)!/7!) \\ G. C. Greubel, Aug 15 2018

a(n)= A051379(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+7)!/7!.
E.g.f.: 1/(1-x)^8.
a(n) = A173333(n+7,7). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+7,n) / 8. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 5040*e - 13699.
Sum_{n>=0} (-1)^n/a(n) = 1855 - 5040/e. (End)

A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

Original entry on oeis.org

1, -1, 2, 2, -3, 3, -6, 8, -6, 4, 24, -30, 20, -10, 5, -120, 144, -90, 40, -15, 6, 720, -840, 504, -210, 70, -21, 7, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10

Offset: 1

Tom Copeland, Feb 25 2014

Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).
The reversed unsigned array is A111492.
See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=
-log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.
Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in  d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.
A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - Wolfdieter Lang, Mar 01 2014
Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - Tom Copeland, Mar 01 2014
Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - Tom Copeland, Mar 02 2014
For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - Tom Copeland, Apr 05 2014
The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - Peter Bala, Feb 13 2017

			The first few row polynomials are
p(1,x)=  1
p(2,x)= -1 + 2x
p(3,x)=  2 - 3x + 3x^2
p(4,x)= -6 + 8x - 6x^2 + 4x^3
p(5,x)= 24 -30x +20x^2 -10x^3 + 5x^4
...........
For n=3: z!/(z-3)!=z^3-3z^2+2z=St1(3,z) with derivative 3z^2-6z+2, and
3·St2(2,x)-6·St2(1,x)+2=3(x^2+x)-6x+2=3x^2-3x+2=p(3,x). To see the relation to the operator formalism, note that (xD)^k=St2(k,:xD:) and (xD)!/(xD-k)!=[St2(·,:xD:)]!/[St2(·,:xD:)-k]!= :xD:^k.
The triangle a(n,k) begins:
n\k       0       1       2      3      4     5      6    7   8   9 ...
1:        1
2:       -1       2
3:        2      -3       3
4:       -6       8      -6      4
5:       24     -30      20    -10      5
6:     -120     144     -90     40    -15     6
7:      720    -840     504   -210     70   -21      7
8:    -5040    5760   -3360   1344   -420   112    -28    8
9:    40320  -45360   25920 -10080   3024  -756    168  -36   9
10: -362880  403200 -226800  86400 -25200  6048  -1260  240 -45  10
... formatted by _Wolfdieter Lang_, Mar 01 2014
-----------------------------------------------------------------------

Tom Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator, 2014.
Tom Copeland, Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering, 2014.
Tom Copeland, Compositional inverse operators and Sheffer sequences, 2016.

Cf. A094587, A132013, A132014, A132159, A235706, A238385.

Cf. A008290, A092271, A211603.

a[n_, k_] := (-1)^(n-k-1)*n!/((n-k)*k!); Table[a[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

a(n,k) = (-1)^(n-k-1)*n!/((n-k)*k!) for k=0 to (n-1).
E.g.f.: log(1+t)*exp(x*t).
E.g.f.for unsigned array: -log(1-t)*exp(x*t).
The lowering op. for the row polynomials is L=d/dx, i.e., L p(n,x) = n*p(n-1,x).
An e.g.f. for an unsigned related version is -log(1+t)*exp(x*t)/t= exp(t*s(·,x)) with s(n,x)=(-1)^n * p(n+1,-x)/(n+1). Let L=d/dx and R= x-(1/((1-D)log(1-D))+1/D),then R s(n,x)= s(n+1,x) and L s(n,x)= n*s(n-1,x), defining a special Sheffer sequence of polynomials, an Appell sequence. So, R (-1)^(n-1) p(n,-x)/n = (-1)^n p(n+1,-x)/(n+1).
From Tom Copeland, Apr 17 2014: (Start)
Dividing each diagonal by its first element (-1)^(n-1)*(n-1)! yields the reverse of A104712.
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x). Then with dP = A132440, M = padded A238363 = A238385-I, I = identity matrix, and (B(.,x))^n = B(n,x) = the n-th Bell polynomial Bell(n,x) of A008277,
A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x), and
B) P(:xD:)=exp(dP:xD:)=exp[(e^M-I):xD:]=exp[M*B(.,:xD:)]=exp[M*xD]=
  (1+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x].
C) P(x)^m = P(m*x). P(2x) = A038207(x) = exp[M*B(.,2x)], face vectors of n-D hypercubes. (End)
From Tom Copeland, Apr 26 2014: (Start)
M = padded A238363 = A238385-I
A)  = [St1]*[dP]*[St2] = [padded A008275]*A132440*A048993
B)  = [St1]*[dP]*[St1]^(-1)
C)  = [St2]^(-1)*[dP]*[St2]
D)  = [St2]^(-1)*[dP]*[St1]^(-1),
  where [St1]=padded A008275 just as [St2]=A048993=padded A008277.
E) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I + dP)^x*[St1].
F) exp(x*M) = [St1]*P(x)*[St2] = (I + dP)^x,
  where (I + dP)^x = sum(k>=0, C(x,k)*dP^k).
Let the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose. Form the power series V(x)= Rv * Cv(x) and W(y) := V(x.) evaluated umbrally with (x.)^n = x_n = (y)_n = y!/(y-n)!. Then
G) U(:xD:) = dV(:xD:)/d(xD) = dW(xD)/d(xD) evaluated with (xD)^n = Bell(n,:xD:),
H) U(x) = dV(x.)/dy := dW(y)/dy evaluated with y^n=y_n=Bell(n,x), and
I) U(x) = Rv * M * Cv(x).   (Cf. A132440, A074909.) (End)
The Bernoulli polynomials Ber_n(x) are related to the polynomials q_n(x) = p(n+1,x) / (n+1) with the e.g.f. [log(1+t)/t] e^(xt) (cf. s_n (x) above) as Ber_n(x) = St2_n[q.(St1.(x))], umbrally, or [St2]*[q]*[St1], in matrix form. Since q_n(x) is an Appell sequence of polynomials, q_n(x) = [log(1+D_x)/D_x]x^n. - Tom Copeland, Nov 06 2016

Pincherle formalism added by Tom Copeland, Feb 27 2014

A049389 a(n) = (n+8)!/8!.

Original entry on oeis.org

1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800, 158789030400, 3016991577600, 60339831552000, 1267136462592000, 27877002177024000, 641171050071552000, 15388105201717248000, 384702630042931200000, 10002268381116211200000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=9) ~ exp(-x)/x*(1 - 9/x + 90/x^2 - 990/x^3 + 11880/x^4 - 154440/x^5 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

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

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A051379, A051380.

Cf. A130534, A163931, A173333, A245334.

a049389 = (flip div 40320) . a000142 . (+ 8)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+8)/40320: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 8)!/8!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

PARI
```
a(n) = (n+8)!/8!;
```

a(n)= A051380(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+8)!/8!.
E.g.f.: 1/(1-x)^9.
a(n) = A173333(n+8,8). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+8,n) / 9. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 40320*e - 109600.
Sum_{n>=0} (-1)^n/a(n) = 40320/e - 14832. (End)

A049398 a(n) = (n+9)!/9!.

Original entry on oeis.org

1, 10, 110, 1320, 17160, 240240, 3603600, 57657600, 980179200, 17643225600, 335221286400, 6704425728000, 140792940288000, 3097444686336000, 71241227785728000, 1709789466857472000, 42744736671436800000, 1111363153457356800000, 30006805143348633600000

Offset: 0

Wolfdieter Lang

The p=9 member of the p-family of sequences {(n+p-1)!/p!}, n >= 1.

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=10) ~ exp(-x)/x*(1 - 10/x + 110/x^2 - 1320/x^3 + 17160/x^4 - 240240/x^5 + 3603600/x^6 - ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

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

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A049389.

Cf. A130534, A163931, A173333, A245334.

a049398 = (flip div 362880) . a000142 . (+ 9)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+9)/362880: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 9)!/9!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

PARI
```
a(n) = (n+9)!/9!
```

E.g.f.: 1/(1-x)^10.
a(n) = A173333(n+9,9). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+9,n) / 10. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 362880*e - 986409.
Sum_{n>=0} (-1)^n/a(n) = 133497 - 362880/e. (End)

cp's OEIS Frontend

A001725 a(n) = n!/5!.

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A052762 Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).

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

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A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

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A033487 a(n) = n*(n+1)*(n+2)*(n+3)/4.

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A001730 a(n) = n!/6!.

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A052762 Products of 4 consecutive integers: a(n) = n(n-1)(n-2)*(n-3).

A033487 a(n) = n(n+1)(n+2)*(n+3)/4.