A124320 -id:A124320 - cp's OEIS frontend

A352601 a(n) = RisingFactorial(2n, n) = A124320(2n, n).

1, 2, 20, 336, 7920, 240240, 8910720, 390700800, 19769460480, 1133836704000, 72684900288000, 5150244363264000, 399703747322880000, 33719008124158156800, 3072176295756632064000, 300649528529562820608000, 31451820032947491201024000, 3502589049123697883750400000

Offset: 0

Peter Luschny, Mar 22 2022

nonn

Cf. A124320.

Table[Pochhammer[2*n, n], {n, 0, 17}] (* Amiram Eldar, Mar 22 2022 *)

from sympy import rf
def A352601(n): return rf(2*n,n) # Chai Wah Wu, Mar 22 2022

def a(n): return rising_factorial(2*n, n)
print([a(n) for n in range(18)])

a(n) ~ 3^(3*n - 1/2) * n^n / (2^(2*n - 1/2) * exp(n)). - Vaclav Kotesovec, Aug 02 2024

A368583 Table read by rows: T(n, k) = A124320(n + 1, k) * A132393(n, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 12, 0, 8, 60, 120, 0, 30, 330, 1260, 1680, 0, 144, 2100, 11760, 30240, 30240, 0, 840, 15344, 113400, 428400, 831600, 665280, 0, 5760, 127008, 1169280, 5821200, 16632000, 25945920, 17297280, 0, 45360, 1176120, 13000680, 80415720, 302702400, 696215520, 908107200, 518918400

Offset: 0

Peter Luschny, Jan 10 2024

			Triangle starts:
  [0] [1]
  [1] [0,   2]
  [2] [0,   3,   12]
  [3] [0,   8,   60,     120]
  [4] [0,  30,   330,   1260,   1680]
  [5] [0, 144,  2100,  11760,  30240,  30240]
  [6] [0, 840, 15344, 113400, 428400, 831600, 665280]

Cf. A124320 (rising factorial), A132393 (unsigned Stirling1), A001813 (main diagonal), A052819 (row sums), A227457 (alternating row sums), A368584.

def Trow(n): return [rising_factorial(n+1, k)*stirling_number1(n, k)
                     for k in range(n+1)]
for n in range(7): print(Trow(n))

A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 12, 0, 4, 60, 120, 0, 5, 210, 1260, 1680, 0, 6, 630, 8400, 30240, 30240, 0, 7, 1736, 45360, 327600, 831600, 665280, 0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280, 0, 9, 11430, 956340, 20207880, 162162000, 575134560, 908107200, 518918400

Offset: 0

Peter Luschny, Jan 10 2024

			Triangle starts:
  [0] [1]
  [1] [0, 2]
  [2] [0, 3,   12]
  [3] [0, 4,   60,    120]
  [4] [0, 5,  210,   1260,    1680]
  [5] [0, 6,  630,   8400,   30240,    30240]
  [6] [0, 7, 1736,  45360,  327600,   831600,   665280]
  [7] [0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280]

Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A124320 (rising factorial), A048993(Stirling2), A053492 (row sums), A213236 (alternating row sums), A001813 (main diagonal), A368583.

def Trow(n): return [rising_factorial(n+1, k)*stirling_number2(n, k) for k in range(n+1)]
for n in range(7): print(Trow(n))

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

A384164 a(n) = Product_{k=0..n-1} (3*n+k).

Original entry on oeis.org

1, 3, 42, 990, 32760, 1395360, 72681840, 4475671200, 318073392000, 25622035084800, 2306992893004800, 229601607198163200, 25028504609870361600, 2965681982933429760000, 379534960108578193920000, 52170410224819317150720000, 7666009844358186506465280000, 1199151678674216896627654656000

Offset: 0

Seiichi Manyama, May 21 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A384164, A384165.
Cf. A000407, A352601.
Cf. A061924, A124320.

[1] cat  [&*[(3*n + k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025

a[n_]:=Product[(3*n+k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)

PARI
```
a(n) = prod(k=0, n-1, 3*n+k);
```

from sympy import rf
def A384164(n): return rf(3*n,n) # Chai Wah Wu, May 21 2025

def a(n): return rising_factorial(3*n, n)

a(n) = RisingFactorial(3*n,n) = A124320(3*n,n) = n! * binomial(4*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n).
a(n) = (3/4) * A061924(n) for n > 0.

A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 2, 1, 0, 1, 10, 3, 7, 3, 0, 1, 15, 4, 25, 12, 2, 0, 1, 21, 5, 65, 30, 6, 1, 0, 1, 28, 6, 140, 60, 12, 15, 7, 0, 1, 36, 7, 266, 105, 20, 90, 50, 12, 0, 1, 45, 8, 462, 168, 30, 350, 195, 60, 6, 0, 1, 55, 9, 750, 252, 42, 1050, 560, 180, 24, 1, 0

Offset: 0

Peter Luschny, Oct 20 2017

			Array starts:
m\j| 0   1  2     3       4       5       6       7       8       9      10
---|-----------------------------------------------------------------------
m=0| 1,  0, 0,    0,      0,      0,      0,      0,      0,      0,      0
m=1| 1,  1, 1,    1,      3,      2,      1,      7,     12,      6,      1
m=2| 1,  3, 2,    7,     12,      6,     15,     50,     60,     24,     31
m=3| 1,  6, 3,   25,     30,     12,     90,    195,    180,     60,    301
m=4| 1, 10, 4,   65,     60,     20,    350,    560,    420,    120,   1701
m=5| 1, 15, 5,  140,    105,     30,   1050,   1330,    840,    210,   6951
m=6| 1, 21, 6,  266,    168,     42,   2646,   2772,   1512,    336,  22827
m=7| 1, 28, 7,  462,    252,     56,   5880,   5250,   2520,    504,  63987
m=8| 1, 36, 8,  750,    360,     72,  11880,   9240,   3960,    720, 159027
m=9| 1, 45, 9, 1155,    495,     90,  22275,  15345,   5940,    990, 359502
   A000217, A001296,A027480,A002378,A001297,A293475,A033486,A007531,A001298
.
m\j| ...      11      12      13      14
---|-----------------------------------------
m=0| ...,      0,      0,      0,      0, ... [A000007]
m=1| ...,     15,     50,     60,     24, ... [A028246]
m=2| ...,    180,    390,    360,    120, ... [A053440]
m=3| ...,   1050,   1680,   1260,    360, ... [A294032]
m=4| ...,   4200,   5320,   3360,    840, ...
m=5| ...,  13230,  13860,   7560,   1680, ...
m=6| ...,  35280,  31500,  15120,   3024, ...
m=7| ...,  83160,  64680,  27720,   5040, ...
m=8| ..., 178200, 122760,  47520,   7920, ...
m=9| ..., 353925, 218790,  77220,  11880, ...
         A293476,A293608,A293615,A052762, ...
.
The parameter m runs over the triangles and j indexes the triangles by reading them by rows. Let T(m, n) denote the row [T(m, n, k) for 0 <= k <= n] and T(m) denote the triangle [T(m, n) for n >= 0]. Then for instance T(2) is the triangle A053440, T(3, 2) is row 2 of A294032 (which is [25, 30, 12]) and T(3, 2, 1) = 30.
.
Remark: To adapt the sequences A028246 and A053440 to our enumeration use the exponential generating functions exp(x)/(1 - y*(exp(x) - 1)) and exp(x)*(2*exp(x) - y*exp(2*x) + 2*y*exp(x) - 1 - y)/(1 - y*(exp(x) - 1))^2 instead of those indicated in their respective entries.

Eric Weisstein's World of Mathematics, Nørlund Polynomial.

A000217(n) = T(n, 1, 0), A001296(n) = T(n, 2, 0), A027480(n) = T(n, 2, 1),
A002378(n) = T(n, 2, 2), A001297(n) = T(n, 3, 0), A293475(n) = T(n, 3, 1),
A033486(n) = T(n, 3, 2), A007531(n) = T(n, 3, 3), A001298(n) = T(n, 4, 0),
A293476(n) = T(n, 4, 1), A293608(n) = T(n, 4, 2), A293615(n) = T(n, 4, 3),
A052762(n) = T(n, 4, 4), A052787(n) = T(n, 5, 5), A000225(n) = T(1, n, 1),
A028243(n) = T(1, n, 2), A028244(n) = T(1, n, 3), A028245(n) = T(1, n, 4),
A032180(n) = T(1, n, 5), A228909(n) = T(1, n, 6), A228910(n) = T(1, n, 7),
A000225(n) = T(2, n, 0), A007820(n) = T(n, n, 0).
A028246(n,k) = T(1, n, k), A053440(n,k) = T(2, n, k), A294032(n,k) = T(3, n, k),
A293926(n,k) = T(n, n, k), A124320(n,k) = T(n, k, k), A156991(n,k) = T(k, n, n).
Cf. A293616.

A293617 := proc(m, n, k) option remember:
if m = 0 then 0^n elif k < 0 or k > n then 0 elif n = 0 then 1 else
(k+m)*A293617(m,n-1,k) + k*A293617(m,n-1,k-1) + A293617(m-1,n,k) fi end:
for m in [$0..4] do for n in [$0..6] do print(seq(A293617(m, n, k), k=0..n)) od od;
# Sample uses:
A027480 := n -> A293617(n, 2, 1): A293608 := n -> A293617(n, 4, 2):
# Flatten:
a := proc(n) local w; w := proc(k) local t, s; t := 1; s := 1;
while t <= k do s := s + 1; t := t + s od; [s - 1, s - t + k] end:
seq(A293617(n - k, w(k)[1], w(k)[2]), k=0..n) end: seq(a(n), n = 0..11);

T[m_, n_, k_] := Pochhammer[m, k] StirlingS2[n + m, k + m];
For[m = 0, m < 7, m++, Print[Table[T[m, n, k], {n,0,6}, {k,0,n}]]]
A293617Row[m_, n_] := Table[T[m, n, k], {k,0,n}];
(* Sample use: *)
A293926Row[n_] := A293617Row[n, n];

T(m,n,k) = (k + m)*T(m, n-1, k) + k*T(m, n-1, k-1) + T(m-1, n, k) with boundary conditions T(0, n, k) = 0^n; T(m, n, k) = 0 if k<0 or k>n; and T(m, 0, k) = 0^k.

T(m,n,k) = Pochhammer(m, k)*binomial(n + m, k + m)*NorlundPolynomial(n - k, -k - m).

A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 15, 24, 1, 1, 1, 5, 28, 105, 120, 1, 1, 1, 6, 45, 280, 945, 720, 1, 1, 1, 7, 66, 585, 3640, 10395, 5040, 1, 1, 1, 8, 91, 1056, 9945, 58240, 135135, 40320, 1, 1, 1, 9, 120, 1729, 22176, 208845, 1106560, 2027025, 362880, 1

Offset: 0

Peter Luschny, Dec 18 2023

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

			Array A(n, k) starts:
  [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
  [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
  [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
  [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
  [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
  [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
  [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
  [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
  [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755

Index entries for sequences related to factorial numbers.

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
Columns: A000012, A000027, A000384, A011199, A011245.
Variant: A256268. Main diagonal: A092985.
Cf. A124320, A008279, A326323.

def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])

Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).

cp's OEIS Frontend

A352601 a(n) = RisingFactorial(2*n, n) = A124320(2*n, n).

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A368583 Table read by rows: T(n, k) = A124320(n + 1, k) * A132393(n, k).

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A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

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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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A384164 a(n) = Product_{k=0..n-1} (3*n+k).

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A293617 Array of triangles read by ascending antidiagonals, T(m, n, k) = Pochhammer(m, k) * Stirling2(n + m, k + m) with m >= 0, n >= 0 and 0 <= k <= n.

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A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

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A352601 a(n) = RisingFactorial(2n, n) = A124320(2n, n).