A265892 -id:A265892 - cp's OEIS frontend

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

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

A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 2, 2, 2, 1, 4, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 4, 1, 1, 0, 1, 1, 3, 1, 1, 6, 3, 1, 5, 1, 1, 0, 1, 1, 4, 1, 1, 1, 8, 4, 1, 5, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 5, 2, 6, 1, 1, 0, 1, 2, 1, 2, 3, 3, 3, 2, 1, 6, 2, 6, 1, 1, 0

Offset: 0

Antti Karttunen, Dec 19 2015

Square array A(row,col) 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), ...

A265609(n,k) is the rising factorial, also known as Pochhammer symbol and A099563(n) is the most significant "digit" (place holder) in the factorial representation (A007623) of n.

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Taking the most significant "digit" (placeholder that may get arbitrarily large values) gives us the top left corner of this array:
-
1, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 2, 2, 3, 3, 4, 4,  5, 5,  6,  6,  7,  7,  8,  8,  9,  9, 10, 10, 11
1, 2, 3, 1, 1, 1, 1, 1,  1, 2,  2,  2,  2,  2,  2,  3,  3,  3,  3,  3,  3
1, 2, 1, 1, 2, 3, 3, 4,  5, 6,  7,  8, 10, 11, 12, 14, 15, 17, 19, 21,  1
1, 1, 1, 2, 4, 6, 8, 1,  1, 1,  1,  2,  2,  2,  2,  3,  3,  3,  4,  4,  5
1, 1, 2, 4, 1, 1, 1, 2,  3, 3,  4,  5,  6,  8,  9, 11, 12, 14, 16, 19, 21
1, 1, 3, 1, 1, 2, 3, 4,  6, 8, 11, 14,  1,  1,  1,  1,  2,  2,  2,  3,  3
1, 1, 3, 1, 2, 3, 5, 8,  1, 1,  1,  2,  2,  3,  4,  5,  6,  7,  8, 10, 12
1, 1, 4, 1, 3, 5, 9, 1,  2, 2,  3,  5,  6,  8, 11, 14, 17, 21,  1,  1,  1
1, 1, 1, 2, 4, 8, 1, 2,  3, 5,  7, 10, 14,  1,  1,  1,  2,  2,  3,  3,  4
1, 2, 1, 3, 6, 1, 2, 4,  6, 9, 14,  1,  1,  2,  3,  4,  5,  6,  8, 10, 13
1, 2, 1, 3, 1, 2, 3, 6, 10, 1,  1,  2,  3,  5,  6,  9, 12, 16, 21,  1,  1
1, 2, 1, 4, 1, 2, 5, 9,  1, 2,  3,  4,  7, 10, 14, 20,  1,  1,  2,  2,  3
1, 2, 2, 5, 1, 3, 7, 1,  2, 3,  5,  8, 13,  1,  1,  1,  2,  3,  4,  6,  8
...

Cf. A007623, A265609.
Column 1: A099563.
Row 0: A000007, rows 1 & 2: A000012, row 3: A008619 (see comment in A001710).
Row 4: 1,2,3 followed by A097992 ?
Main diagonal: A265891 (essentially, without the initial 1 from the corner of this array).
Cf. also array A265892.

(define (A265890 n) (A265890bi (A025581 n) (A002262 n)))
(define (A265890bi row col) (A099563 (A265609bi row col))) ;; Code for A265609bi given in A265609.

A265893 a(n) = A084558(n) - A230403(n); the length of factorial base representation of n without its trailing zeros.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 1

Offset: 0

Antti Karttunen, Dec 20 2015

			In factorial base A007623, 0 is shown as "0", but in this case all the zeros are trailing, so we set a(0) = 0 by convention.
For n = 2, A007623(2) = "10", and by discarding the trailing zero only one significant digit "1" is left, thus a(2) = 1.
For n = 132, A007623(132) = "10200", and by discarding its trailing zeros we are left with just three digits "102", thus a(132) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation.

Cf. A007623, A084558, A230403.

Column 1 of A265892.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Length[s] - FirstPosition[s, ?(#>0 &)][[1]] + 1]; a[0] = 0; Array[a, 100, 0] (* _Amiram Eldar, Feb 21 2024 *)

(define (A265893 n) (- (A084558 n) (A230403 n)))

a(n) = A084558(n) - A230403(n).

cp's OEIS Frontend

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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A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

A265893 a(n) = A084558(n) - A230403(n); the length of factorial base representation of n without its trailing zeros.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula