A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.
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, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1
Offset: 0
Examples
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, 5, 12, 21, 32, 45, 60, 77, 96, ... [3] 0, 45, 120, 231, 384, 585, 840, 1155, 1536, ... [4] 0, 585, 1680, 3465, 6144, 9945, 15120, 21945, 30720, ... [5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ... . Seen as the triangle T(n, k) = A(n - k, k): [0] 1; [1] 0, 1; [2] 0, 1, 1; [3] 0, 5, 2, 1; [4] 0, 45, 12, 3, 1; [5] 0, 585, 120, 21, 4, 1; [6] 0, 9945, 1680, 231, 32, 5, 1; [7] 0, 208845, 30240, 3465, 384, 45, 6, 1;
Crossrefs
Programs
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Maple
A := (n, k) -> 4^n*pochhammer(k/4, n): for n from 0 to 5 do seq(A(n, k), k = 0..9) od; T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9); # Using the exponential generating functions of the columns: EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4); ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end: seq(lprint(EGFcol(n, 9)), n = 0..5); # Using the generating polynomials for the rows: P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n): seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5); # Implementing the LU decomposition of A: with(LinearAlgebra): L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)): U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)): MatrixMatrixMultiply(L, Transpose(U));
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Mathematica
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 *) -
SageMath
def A(n, k): return 4**n * rising_factorial(k/4, n) for n in range(6): print([A(n, k) for k in range(9)])
Formula
A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).
A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.
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).
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).
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.
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