A159083 -id:A159083 - cp's OEIS frontend

A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Reinhard Zumkeller, Feb 19 2010

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Index entries for sequences related to factorial numbers.

Row sums give A002627.
Central terms give A006963:
T(2*n-1,n) = A006963(n+1).
T(2*n,n) = A001813(n).
T(2*n,n+1) = A001761(n).
1 < k <= n: T(n,k) = T(n,k-1) / k.
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).
1 <= k <= n: T(n+1,k+1) = A162995(n,k).
T(n,1) = A000142(n).
T(n,2) = A001710(n) for n>1.
T(n,3) = A001715(n) for n>2.
T(n,4) = A001720(n) for n>3.
T(n,5) = A001725(n) for n>4.
T(n,6) = A001730(n) for n>5.
T(n,7) = A049388(n-7) for n>6.
T(n,8) = A049389(n-8) for n>7.
T(n,9) = A049398(n-9) for n>8.
T(n,10) = A051431(n) for n>9.
T(n,n-7) = A159083(n+1) for n>7.
T(n,n-6) = A053625(n+1) for n>6.
T(n,n-5) = A052787(n) for n>5.
T(n,n-4) = A052762(n) for n>4.
T(n,n-3) = A007531(n) for n>3.
T(n,n-2) = A002378(n-1) for n>2.
T(n,n-1) = A000027(n) for n>1.
T(n,n) = A000012(n).
Cf. A138533, A002627.

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

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

A239035 Product of 8 consecutive integers. a(n) = RisingFactorial(n, 8).

Original entry on oeis.org

0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200, 518918400, 980179200, 1764322560, 3047466240, 5079110400, 8204716800, 12893126400, 19769460480, 29654190720, 43609104000, 62990928000, 89513424000, 125318793600, 173059286400

Offset: 0

Michel Marcus, Mar 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vincent Thill, Nombres consécutifs et carrés, author's blog, Aug 03 2013 (consecutive numbers and squares).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. Product of n consecutive integers: A002378 (n=2), A007531 (n=3), A052762 (n=4), A052787 (n=5), A053625 (n=6), A159083 (n=7).

[(n^4+14*n^3+63*n^2+98*n+28)^2-(28+8*n)^2: n in [0..30]]; // Vincenzo Librandi, Mar 11 2014

CoefficientList[Series[40320 x/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Mar 11 2014 *)

PARI
```
a(n) = prod(k=0, 7, n+k);
```

concat([0], Vec(40320*x/(1-x)^9  + O(x^100))) \\ Colin Barker, Mar 09 2014

print([rising_factorial(n, 8) for n in range(23)]) # Peter Luschny, Mar 22 2022

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7).
a(n) = (n^4+14*n^3+63*n^2+98*n+28)^2 - (28+8*n)^2. (see Thill link).
G.f.: 40320*x / (1-x)^9. - Colin Barker, Mar 09 2014
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/35280.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2)/315 - 1163/66150. (End)

A162995 A scaled version of triangle A162990.

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 60, 20, 5, 1, 360, 120, 30, 6, 1, 2520, 840, 210, 42, 7, 1, 20160, 6720, 1680, 336, 56, 8, 1, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Johannes W. Meijer, Jul 27 2009

We get this scaled version of triangle A162990 by dividing the coefficients in the left hand columns by their 'top-values' and then taking the square root.
T(n,k) = A173333(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Feb 19 2010
T(n,k) = A094587(n+1,k+1), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

			The first few rows of the triangle are:
[1]
[3, 1]
[12, 4, 1]
[60, 20, 5, 1]

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A094587.
A056542(n) equals the row sums for n>=1.
A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431 are related to the left hand columns.
A000012, A009056, A002378, A007531, A052762, A052787, A053625 and A159083 are related to the right hand columns.

a162995 n k = a162995_tabl !! (n-1) !! (k-1)
a162995_row n = a162995_tabl !! (n-1)
a162995_tabl = map fst $ iterate f ([1], 3)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

a := proc(n, m): (n+1)!/(m+1)! end: seq(seq(a(n, m), m=1..n), n=1..9); # Johannes W. Meijer, revised Nov 23 2012

Table[(n+1)!/(m+1)!, {n, 10}, {m, n}] (* Paolo Xausa, Mar 31 2024 *)

a(n,m) = (n+1)!/(m+1)! for n = 1, 2, 3, ..., and m = 1, 2, ..., n.

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A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

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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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A239035 Product of 8 consecutive integers. a(n) = RisingFactorial(n, 8).

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A162995 A scaled version of triangle A162990.

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