A173333 -id:A173333 - 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

A006963 Number of planar embedded labeled trees with n nodes: (2*n-3)!/(n-1)! for n >= 2, a(1) = 1.

Original entry on oeis.org

1, 1, 3, 20, 210, 3024, 55440, 1235520, 32432400, 980179200, 33522128640, 1279935820800, 53970627110400, 2490952020480000, 124903451312640000, 6761440164390912000, 393008709555221760000, 24412776311194951680000, 1613955767240110694400000

Offset: 1

Simon Plouffe and N. J. A. Sloane

For n>1: central terms of the triangle in A173333; cf. A001761, A001813. - Reinhard Zumkeller, Feb 19 2010
Can be obtained from the Vandermonde permanent of the first n positive integers; see A093883. - Clark Kimberling, Jan 02 2012
All trees can be embedded in the plane, but "planar embedded" means that orientation matters but rotation doesn't. For example, the n-star with n-1 edges has n! ways to label it, but rotation removes a factor of n-1. Another example, the n-path has n! ways to label it, but rotation removes a factor of 2. - Michael Somos, Aug 19 2014

			G.f. = x + x^2 + 3*x^3 + 20*x^4 + 210*x^5 + 3024*x^6 + 55440*x^7 + 1235520*x^8 + ...
a(5) = 210 = 30 + 60 + 120 where 30 is for the star, 60 for the path, and 120 for the tree with one trivalent vertex. - _Michael Somos_, Aug 19 2014

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 = 1..200
David Callan, A quick count of plane (or planar embedded) labeled trees.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, Journal of Knot Theory and Its Ramifications, Vol. 25, No. 8 (2016), 1650047; arXiv preprint, arXiv:1507.03163 [math.CO], 2015-2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 109.
Bradley Robert Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80.
Pierre Leroux and Brahim Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80. (Annotated scanned copy)
J. W. Moon, Counting Labelled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.
Ran J. Tessler, A Cayley-type identity for trees, arXiv:1809.00001 [math.CO], 2018.
Index entries for sequences related to trees.

Cf. A000407, A001761, A001813, A173333.

[1] cat [Factorial(2*n-3)/Factorial(n-1): n in [2..20]]; // Vincenzo Librandi, Nov 12 2011

1, seq((2*n-3)!/(n-1)!,n=2..30); # Robert Israel, Aug 20 2014

Join[{1},Table[(2n-3)!/(n-1)!,{n,2,20}]] (* Harvey P. Dale, Nov 03 2011 *)
a[ n_] := With[{m = n - 1}, If[m < 1, Boole[m == 0], m! SeriesCoefficient[ -Log[(1 + Sqrt[1 - 4 x]) / 2], {x, 0, m}]]] (* Michael Somos, Jul 01 2013 *)
a[ n_] := If[n < 2, Boole[n == 1], (2 n - 3)! / (n - 1)!]; (* Michael Somos, Aug 19 2014 *)
a[1] := 1; a[n_] := (-1)^(n - 1)*Sum[(-1)^k*Binomial[2*n - 3, n + k - 2]*StirlingS1[n + k - 1, k + 1], {k, 1, n - 1}]; Flatten[Table[a[n], {n, 1, 19}]] (* Detlef Meya, Jan 18 2024 *)

{a(n) = n--; if( n<1, n==0, n! * polcoeff( -log( (1 + sqrt(1 - 4*x + x * O(x^n))) / 2), n))}; /* Michael Somos, Jul 01 2013 */

def A006963(n): return 1 if n==1 else factorial(2*n-3)/factorial(n-1)
[A006963(n) for n in range(1,31)] # G. C. Greubel, May 23 2023

E.g.f. for a(n+1), n >= 1, log(c(x)); c(x) = g.f. for Catalan numbers A000108. - Wolfdieter Lang
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n) = int(x^n * erfc(sqrt(x)/2)/2, x=0..infinity), n=0, 1..., where erfc(x) is the complementary error function. - Karol A. Penson, Sep 27 2001
a(n) ~ 2^(-5/2)*n^-2*2^(2*n)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
a(n+1) = (n+1)*(n+2)*...*(2n-1) for n>=2. - Jaroslav Krizek, Nov 09 2010
E.g.f. (A(x)-1) is reversion of exp(-x)-exp(-2*x). - Vladimir Kruchinin, Jan 30 2012
G.f.: 1 + x*G(0) where G(k) = 1 + x*(2*k+1)*(4*k+3)/(k + 1 - 4*x*(k+1)^2*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
E.g.f.: 1 + x*E(0) where E(k) = 1 + x*(2*k+1)*(4*k+3)/(2*(k + 1)^2 - 8*x*(k+1)^3*(4*k+5)/(4*x*(k+1)*(4*k+5) + (2*k+3)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 02 2013
E.g.f: sqrt(1-4*x)/4 - 1/4 + 3*x/2 - x*log((1+sqrt(1-4*x))/2). - Robert Israel, Aug 20 2014
D-finite with recurrence (-n+1)*a(n) +2*(2*n-3)*(n-2)*a(n-1)=0. - R. J. Mathar, Jan 03 2018
From Amiram Eldar, Apr 03 2022: (Start)
Sum_{n>=1} 1/a(n) = 3/2 + 3*exp(1/4)*sqrt(Pi)*erf(1/2)/4, where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/2 - sqrt(Pi)*erfi(1/2)/(4*exp(1/4)), where erfi is the imaginary error function. (End)
a(n) = A000407(n-2)/(n-1). - R. J. Mathar, Mar 30 2023
a(1) = 1; a(n) = (-1)^(n - 1)*Sum_{k=1..n - 1} (-1)^k*binomial(2*n - 3, n + k - 2)*Stirling1(n + k - 1, k + 1). - Detlef Meya, Jan 18 2024

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)

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)

A052787 Product of 5 consecutive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 720, 2520, 6720, 15120, 30240, 55440, 95040, 154440, 240240, 360360, 524160, 742560, 1028160, 1395360, 1860480, 2441880, 3160080, 4037880, 5100480, 6375600, 7893600, 9687600, 11793600, 14250600, 17100720, 20389320, 24165120, 28480320, 33390720

Offset: 0

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

Appears in Harriot along with the formula (for a different offset) a(n) = n^5 + 10n^4 + 35n^3 + 50n^2 + 24n, see links. - Charles R Greathouse IV, Oct 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Thomas Harriot, Manuscript 6782, p. 77, c. 1599.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 744.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002378, A007531, A052762, A000389, A054559, A173333.

[n*(n-1)*(n-2)*(n-3)*(n-4): n in [0..35]]; // Vincenzo Librandi, May 26 2011

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,5), n=0..31); # Zerinvary Lajos, Apr 26 2007
G(x):=x^5*exp(x): f[0]:=G(x): for n from 1 to 31 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..31); # Zerinvary Lajos, Apr 05 2009

Times@@@(Partition[Range[-4,35],5,1])  (* Harvey P. Dale, Feb 04 2011 *)

a(n)=120*binomial(n,5) \\ Charles R Greathouse IV, Nov 20 2011

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)=n!/(n-5)!. [Corrected by Philippe Deléham, Dec 12 2003]
a(n) = 120*A000389(n) = 4*A054559(n).
E.g.f.: x^5*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-1-n)*a(n)+(-4+n)*a(n+1), a(5)=120}.
O.g.f.: 120*x^5/(-1+x)^6. - R. J. Mathar, Nov 16 2007
For n>5: a(n) = A173333(n,n-5). - Reinhard Zumkeller, Feb 19 2010
a(n) = a(n-1) + 5*A052762(n). - J. M. Bergot, May 30 2012
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/96.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*log(2)/3 - 131/288. (End)

More terms from Henry Bottomley, Mar 20 2000

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)

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset: 0

Peter Luschny, Dec 19 2015

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
From Antti Karttunen, Dec 19 2015: (Start)
Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

			Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
Index entries for sequences related to factorial numbers

Triangle giving terms only up to column k=n: A124320.
Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
Row sums of the triangle: A000522.
A(n, n) = A000407(n-1) for n>0.
2^n*A(1/2,n) = A001147(n).
Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])

(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015
E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018
A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022
G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

A051431 a(n) = (n+10)!/10!.

Original entry on oeis.org

1, 11, 132, 1716, 24024, 360360, 5765760, 98017920, 1764322560, 33522128640, 670442572800, 14079294028800, 309744468633600, 7124122778572800, 170978946685747200, 4274473667143680000, 111136315345735680000, 3000680514334863360000, 84019054401376174080000

Offset: 0

Wolfdieter Lang

The p=10 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=11) ~ exp(-x)/x*(1 - 11/x + 132/x^2 - 1716/x^3 + 24024/x^4 - 360360/x^5 + 5765760/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, A049398.

Cf. A130534, A163931, A173333, A245334.

a051431 = (flip div 3628800) . a000142 . (+ 10)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+10)/3628800: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 10)!/10!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

a(n) = (n+10)!/10!
E.g.f.: 1/(1-x)^11.
a(n) = A173333(n+10,10). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+10,n) / 11. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 3628800*e - 9864100.
Sum_{n>=0} (-1)^n/a(n) = 3628800/e - 1334960. (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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A006963 Number of planar embedded labeled trees with n nodes: (2*n-3)!/(n-1)! for n >= 2, a(1) = 1.

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

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

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

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