A370915 - OEIS

A370915

A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

%I #25 Mar 20 2024 16:58:37

%S 1,0,1,0,1,1,0,5,2,1,0,45,12,3,1,0,585,120,21,4,1,0,9945,1680,231,32,

%T 5,1,0,208845,30240,3465,384,45,6,1,0,5221125,665280,65835,6144,585,

%U 60,7,1,0,151412625,17297280,1514205,122880,9945,840,77,8,1

%N A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

%C The sequence of square arrays A(m, n, k) starts: A094587 (m = 1), A370419 (m = 2), A371077(m = 3), this array (m = 4).

%F A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).

%F A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.

%F The exponential generating function for column k is (1 - 4*x)^(-k/4). But much more is true: (1 - m*x)^(-k/m) are the exponential generating functions for the columns of the arrays A(m, n, k) = m^n*Pochhammer(k/m, n).

%F The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-4)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).

%F In A370419 _Werner Schulte_ pointed out how A371025 is related to the LU decomposition of A370419. Here the same procedure can be used and amounts to A = A371026 * transpose(binomial triangle), where '*' denotes matrix multiplication. See the Maple section for an implementation.

%e The array starts:

%e [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ...

%e [2] 0, 5, 12, 21, 32, 45, 60, 77, 96, ...

%e [3] 0, 45, 120, 231, 384, 585, 840, 1155, 1536, ...

%e [4] 0, 585, 1680, 3465, 6144, 9945, 15120, 21945, 30720, ...

%e [5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...

%e .

%e Seen as the triangle T(n, k) = A(n - k, k):

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 1, 1;

%e [3] 0, 5, 2, 1;

%e [4] 0, 45, 12, 3, 1;

%e [5] 0, 585, 120, 21, 4, 1;

%e [6] 0, 9945, 1680, 231, 32, 5, 1;

%e [7] 0, 208845, 30240, 3465, 384, 45, 6, 1;

%p A := (n, k) -> 4^n*pochhammer(k/4, n):

%p for n from 0 to 5 do seq(A(n, k), k = 0..9) od;

%p T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);

%p # Using the exponential generating functions of the columns:

%p EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);

%p ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:

%p seq(lprint(EGFcol(n, 9)), n = 0..5);

%p # Using the generating polynomials for the rows:

%p P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):

%p seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);

%p # Implementing the LU decomposition of A:

%p with(LinearAlgebra):

%p L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):

%p U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):

%p MatrixMatrixMultiply(L, Transpose(U));

%t A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Mar 06 2024 *)

%o (SageMath)

%o def A(n, k): return 4**n * rising_factorial(k/4, n)

%o for n in range(6): print([A(n, k) for k in range(9)])

%Y Similar square arrays: A094587, A370419, A371077.

%Y Rows: A000012, A001477, A028347, A370914.

%Y Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.

%Y Cf. A370913 (row sums of triangle), A371026.

%K nonn,tabl,easy

%O 0,8

%A _Peter Luschny_, Mar 06 2024