A000407 -id:A000407 - cp's OEIS frontend

A034177 a(n) is the n-th quartic factorial number divided by 4.

1, 8, 96, 1536, 30720, 737280, 20643840, 660602880, 23781703680, 951268147200, 41855798476800, 2009078326886400, 104472072998092800, 5850436087893196800, 351026165273591808000, 22465674577509875712000, 1527665871270671548416000, 109991942731488351485952000

Offset: 1

Wolfdieter Lang

			G.f. = x + 8*x^2 + 96*x^3 + 1536*x^4 + 30720*x^5 + 737820*x^6 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..365
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 513.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A007696, A000407, A034176. First column of triangle A048786.
A052570 is an essentially identical sequence. - Philippe Deléham, Sep 18 2008
Equals the second right hand column of A167569 divided by 2. - Johannes W. Meijer, Nov 12 2009

List([1..20], n-> 4^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 15 2019

[4^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019

[seq(n!*4^(n-1), n=1..16)]; # Zerinvary Lajos, Sep 23 2006

Array[4^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 4^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019

[4^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

4*a(n) = (4*n)(!^4) = Product_{j=1..n} 4*j = 4^n * n!.
E.g.f.: (-1 + 1/(1-4*x))/4.
D-finite with recurrence: a(n) -4*n*a(n-1)=0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(exp(1/4)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(1-exp(-1/4)). (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

A081125 a(n) = n! / floor(n/2)!.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 120, 840, 1680, 15120, 30240, 332640, 665280, 8648640, 17297280, 259459200, 518918400, 8821612800, 17643225600, 335221286400, 670442572800, 14079294028800, 28158588057600, 647647525324800, 1295295050649600

Offset: 0

Paul Barry, Mar 07 2003

Product of the largest parts in the partitions of n+1 into exactly two parts, n > 0. - Wesley Ivan Hurt, Jan 26 2013 (Clarified on Apr 20 2016)

			a(3) = 6 since 3+1 = 4 has two partitions into two parts, (3,1) and (2,2), and the product of the largest parts is 6. - _Wesley Ivan Hurt_, Jan 26 2013 (Clarified on Apr 20 2016)

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Index to divisibility sequences.

Cf. A004526, A056040, A081123, A000407 (bisection), A001813 (bisection).

[Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // Vincenzo Librandi, Sep 13 2011

Method 1)  a:=n->n!/floor(n/2)!; seq(a(k),k=0..40); # Wesley Ivan Hurt, Jun 03 2013
Method 2)  with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # Wesley Ivan Hurt, Jun 06 2013

Table[n!/Floor[n/2]!, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 20 2016 *)

a(n)=n!/(n\2)! \\ Charles R Greathouse IV, Sep 13 2011

from sympy import rf
def A081125(n): return rf((m:=n+1>>1)+(n+1&1),m) # Chai Wah Wu, Jul 22 2022

def a(n): return rising_factorial(ceil(n/2),floor(n/2))
[a(n) for n in range(26)]  # Peter Luschny, Oct 09 2013

E.g.f.: (1+x)*exp(x^2). - Vladeta Jovovic, Sep 24 2003
From Peter Luschny, Aug 07 2009: (Start)
a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial (A056040).
a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
E.g.f.: E(0) where E(k) = 1 + x/(1 - x/(x + (k+1)/E(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
G.f.: G(0) where G(k) =  1 + x*(2*k+1)/(1 - 2*x/(2*x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
D-finite with recurrence a(n) +2*a(n-1) -2*n*a(n-2) +4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
From Wesley Ivan Hurt, Jun 06 2013: (Start)
a(n) = n!/(n-floor((n+1)/2))!.
a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial (A056040). - Peter Luschny, Oct 28 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
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).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 9, 117, 1989, 41769, 1044225, 30282525, 999323325, 36974963025, 1515973484025, 68218806781125, 3342721532275125, 177164241210581625, 10098361749003152625, 616000066689192310125, 40040004334797500158125

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1) (rows m=0..4).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(9/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 8, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(9/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(9/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+5)(!^4))/5(!^4).

E.g.f.: 1/(1-4*x)^(9/4).

A257621 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(n) = 4*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 42, 9, 27, 393, 393, 27, 81, 3156, 8646, 3156, 81, 243, 23631, 142446, 142446, 23631, 243, 729, 171006, 2015895, 4273380, 2015895, 171006, 729, 2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187, 6561, 8584872, 320039388, 2136524184, 3891302790, 2136524184, 320039388, 8584872, 6561

Offset: 0

Dale Gerdemann, May 09 2015

			Array t(n,k) begins as:
    1,       3,         9,          27,            81, ...;
    3,      42,       393,        3156,         23631, ...;
    9,     393,      8646,      142446,       2015895, ...;
   27,    3156,    142446,     4273380,     102402705, ...;
   81,   23631,   2015895,   102402705,    3891302790, ...;
  243,  171006,  26107983,  2136524184,  123074809242, ...;
  729, 1216725, 320039388, 40688926236, 3437022383970, ...;
Triangle T(n,k) begins as:
     1;
     3,       3;
     9,      42,        9;
    27,     393,      393,        27;
    81,    3156,     8646,      3156,        81;
   243,   23631,   142446,    142446,     23631,      243;
   729,  171006,  2015895,   4273380,   2015895,   171006,     729;
  2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000407 (row sums), A142459, A257612.
Cf. A038221, A257180, A257611, A257620, A257623, A257625, A257627.
Similar sequences listed in A256890.

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,4,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257621(n,k): return t(n-k,k,4,3)
flatten([[A257621(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.
Sum_{k=0..n} T(n, k) = A000407(n).
From G. C. Greubel, Mar 01 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)

A051618 a(n) = (4*n+6)(!^4)/6(!^4).

Original entry on oeis.org

1, 10, 140, 2520, 55440, 1441440, 43243200, 1470268800, 55870214400, 2346549004800, 107941254220800, 5397062711040000, 291441386396160000, 16903600410977280000, 1048023225480591360000, 69169532881719029760000, 4841867301720332083200000, 358298180327304574156800000

Offset: 0

Wolfdieter Lang

This sequence is related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).
Row m=6 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.
a(n) = A001813 a(n+2)/12. - Zerinvary Lajos, Feb 15 2008
For n>4, a(n) mod n^2 = n*(n-2) if n is prime, otherwise 0. - Gary Detlefs, Apr 16 2012

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617 through A051622 (rows m=0..10).

[Factorial(2*n+4)/(12*Factorial(n+2)): n in [0..100]]; // Vincenzo Librandi, Jul 04 2015

seq(mul((n+2+k), k=1..n+2)/12, n=0..17); # Zerinvary Lajos, Feb 15 2008
A051618 := n -> 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3:
seq(simplify(A051618(n)), n = 0..19);  # Peter Luschny, Jan 22 2025

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 9, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
f[n_] := (2n + 4)!/(12(n + 2)!); Array[f, 16, 0] (* Or *)
FoldList[ #2*#1 &, 1, Range[10, 66, 4]] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[1/(1-4x)^(5/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 24 2015 *)
Table[(Product[(4*k + 6), {k, 0, n}])/6, {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)

A051618(n):=(2*n+4)!/(12*(n+2)!)$
makelist(A051618(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

for(n=0,25, print1((2*n+3)!/(6*(n+1)!), ", ")) \\ G. C. Greubel, Jan 27 2017

a(n) = ((4*n+6)(!^4))/6(!^4).
E.g.f.: 1/(1-4*x)^(5/2).
a(n) = (2n+4)!/(12(n+2)!). -  Gary Detlefs, Mar 06 2011
a(n) = (2*n+3)!/(6*(n+1)!). - Gary Detlefs, Apr 16 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = (4^(1+n)*Gamma(5/2+n))/(3*sqrt(Pi)). - Gerry Martens, Jul 02 2015
a(n) ~ 2^(2*n+5/2) * n^(n+2) / (3*exp(n)). - Vaclav Kotesovec, Jul 04 2015
a(n) = 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3. - Peter Luschny, Jan 22 2025

A124320 Triangle read by rows: T(n,k) = k!*binomial(n+k-1,k) (n >= 0, 0 <= k <= n), rising factorial power, Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 1, 3, 12, 60, 1, 4, 20, 120, 840, 1, 5, 30, 210, 1680, 15120, 1, 6, 42, 336, 3024, 30240, 332640, 1, 7, 56, 504, 5040, 55440, 665280, 8648640, 1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800

Offset: 0

Emeric Deutsch, Oct 26 2006

This is the Pochhammer function which is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1. Also known as the rising factorial power. - Peter Luschny, Jan 09 2011

			Triangle starts:
[0]  1;
[1]  1, 1;
[2]  1, 2,   6;
[3]  1, 3,  12,  60;
[4]  1, 4,  20, 120,  840;
[5]  1, 5,  30, 210, 1680, 15120;
[6]  1, 6,  42, 336, 3024, 30240, 332640;
[7]  1, 7,  56, 504, 5040, 55440, 665280, 8648640;
Array starts:
[0] 1,  1,   6,   60,   840,   15120,   332640,   8648640, ... A000407
[1] 1,  2,  12,  120,  1680,   30240,   665280,  17297280, ... A001813
[2] 1,  3,  20,  210,  3024,   55440,  1235520,  32432400, ... A006963
[3] 1,  4,  30,  336,  5040,   95040,  2162160,  57657600, ... A001761
[4] 1,  5,  42,  504,  7920,  154440,  3603600,  98017920, ... A102693
[5] 1,  6,  56,  720, 11880,  240240,  5765760, 160392960, ... A093197
[6] 1,  7,  72,  990, 17160,  360360,  8910720, 253955520, ... A203473
[7] 1,  8,  90, 1320, 24024,  524160, 13366080, 390700800, ...
[8] 1,  9, 110, 1716, 32760,  742560, 19535040, 586051200, ...
[9] 1, 10, 132, 2184, 43680, 1028160, 27907200, 859541760, ...

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
NIST Digital Library of Mathematical Functions, Pochhammer's Symbol

Rows of array: A000407, A001813, A006963, A001761, A102693, A093197, A203473.

Cf. A123680 (row sums), A352601 (array main diagonal), A123680, A068424 (falling factorial power).

T:=proc(n,k) if k<=n then binomial(n+k-1,k)*k! else 0 fi end: for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
A124320 := (n,k)-> `if`(n=0 and k=0,1,pochhammer(n,k)); seq(print(seq(A124320(n,k),k=0..n)),n=0..5); # Peter Luschny, Jan 09 2011

Table[Pochhammer[n,k], {n,0,5},{k,0,n}]//Flatten (* Peter Luschny, Jan 09 2011 *)

for(n=0,10, for(k=0,n, print1(if(n==0 && k==0, 1, (n+k-1)!/(n-1)!), ", "))) \\ G. C. Greubel, Nov 19 2017

for n in (0..5) : [rising_factorial(n, k) for k in (0..n)] # Peter Luschny, Jan 09 2011

T(n,k) = GAMMA(n+k)/GAMMA(n) for n>0. - Peter Luschny, Jan 09 2011

A303486 a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).

Original entry on oeis.org

1, 1, 10, 162, 3640, 104720, 3674160, 152152000, 7264216960, 392841187200, 23734494784000, 1584471003315200, 115825295634048000, 9201578813819392000, 789383453851632640000, 72728093032166347776000, 7162140885524461957120000, 750766815289210771251200000

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*5 = 10;
a(3) = 3*6*9 = 162;
a(4) = 4*7*10*13 = 3640;
a(5) = 5*8*11*14*17 = 104720, etc.

G. C. Greubel, Table of n, a(n) for n = 0..343
Index entries for sequences related to factorial numbers

Column k=3 of A303489.

Cf. A000407, A007559, A008544, A032031, A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609, A113551, A303487, A303488.

Table[n! SeriesCoefficient[1/(1 - 3 x)^(n/3), {x, 0, n}], {n, 0, 17}]
Table[Product[3 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[3^n Pochhammer[n/3, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (3*k + n).
a(n) = 3^n*Gamma(4*n/3)/Gamma(n/3).
a(n) ~ 2^(8*n/3-1)*n^n/exp(n).

A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

Original entry on oeis.org

1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.

Index entries for sequences related to factorial numbers

Column k=4 of A303489.

Cf. A000407, A001813, A007696, A008545, A034176, A034177, A047053, A051617, A051618, A051619, A051620, A051621, A051622, A113551, A303486, A303488.

Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (4*k + n).
a(n) = 4^n*Gamma(5*n/4)/Gamma(n/4).
a(n) ~ 5^(5*n/4-1/2)*n^n/exp(n).

cp's OEIS Frontend

A034177 a(n) is the n-th quartic factorial number divided by 4.

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

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A081125 a(n) = n! / floor(n/2)!.

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A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

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A051617 a(n) = (4*n+5)(!^4)/5(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).

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A257621 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.

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A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

A257621 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), and f(n) = 4*n + 3.