A371076 -id:A371076 - cp's OEIS frontend

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 28, 10, 3, 1, 0, 280, 80, 18, 4, 1, 0, 3640, 880, 162, 28, 5, 1, 0, 58240, 12320, 1944, 280, 40, 6, 1, 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1, 0, 24344320, 4188800, 524880, 58240, 6160, 648, 70, 8, 1

Offset: 0

Werner Schulte and Peter Luschny, Mar 10 2024

			The array starts:
  [0] 1,    1,     1,     1,     1,      1,      1,      1,      1, ...
  [1] 0,    1,     2,     3,     4,      5,      6,      7,      8, ...
  [2] 0,    4,    10,    18,    28,     40,     54,     70,     88, ...
  [3] 0,   28,    80,   162,   280,    440,    648,    910,   1232, ...
  [4] 0,  280,   880,  1944,  3640,   6160,   9720,  14560,  20944, ...
  [5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
  [0] 1;
  [1] 0,       1;
  [2] 0,       1,      1;
  [3] 0,       4,      2,     1;
  [4] 0,      28,     10,     3,    1;
  [5] 0,     280,     80,    18,    4,   1;
  [6] 0,    3640,    880,   162,   28,   5,  1;
  [7] 0,   58240,  12320,  1944,  280,  40,  6, 1;
  [8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
.
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1    1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2    3    4 ... |
    | 0   4   2        | * |     1 3 6 ... | = | 0   4  10   18   28 ... |
    | 0  28  24   6    |   |       1 4 ... |   | 0  28  80  162  280 ... |
    | 0 280 320 144 24 |   |         1 ... |   | 0 280 880 1944 3640 ... |
    | . . .            |   | . . .         |   | . . .                   |

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Family m^n*Pochhammer(k/m, n): A094587 (m=1), A370419 (m=2), this sequence (m=3), A370915 (m=4).
Rows: A000012, A001477, A028552.
Columns: A000007, A007559, A008544, A032031, A007559 (shifted), A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609.
Cf. A303486 (main diagonal), A371079 (row sums of triangle), A371076, A371080.

A := (n, k) -> 3^n*pochhammer(k/3, n):
A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
# Read by antidiagonals:
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
# Using the generating polynomials of the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint([k], EGFcol(k, 8)), k = 0..6);
# As a matrix product:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371076(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));

Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 14 2024 *)

def A(n, k): return 3**n * rising_factorial(k/3, n)
def A(n, k): return (-3)**n * falling_factorial(-k/3, n)

A(n, k) = Product_{j=0..n-1} (3*j + k).
A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
E.g.f. of column k: (1 - 3*t)^(-k/3).
E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371076, i.e., A(n, k) = Sum_{i=0..k} A371076(n, i) * binomial(k, i).

A371080 Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1

Offset: 0

Peter Luschny, Mar 12 2024

			Triangle starts:
[0] 1;
[1] 0,       1;
[2] 0,       4,      1;
[3] 0,      28,     12,      1;
[4] 0,     280,    160,     24,     1;
[5] 0,    3640,   2520,    520,    40,    1;
[6] 0,   58240,  46480,  11880,  1280,   60,  1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;

Peter Luschny, The Bell transform.

Variant: A035469.
Cf. A264428, A371076, A371077.
Cf. A007559, A144346.
Cf. A035342, A049029, A271703.

a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 13 2024

T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = A371076(n, k) / k!.
From Werner Schulte, Mar 13 2024: (Start)
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
From Seiichi Manyama, Apr 19 2025: (Start)
T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)

cp's OEIS Frontend

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

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