A370419 A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.
1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 8, 3, 1, 0, 105, 48, 15, 4, 1, 0, 945, 384, 105, 24, 5, 1, 0, 10395, 3840, 945, 192, 35, 6, 1, 0, 135135, 46080, 10395, 1920, 315, 48, 7, 1, 0, 2027025, 645120, 135135, 23040, 3465, 480, 63, 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, 3, 8, 15, 24, 35, 48, 63, 80, ...
[3] 0, 15, 48, 105, 192, 315, 480, 693, 960, ...
[4] 0, 105, 384, 945, 1920, 3465, 5760, 9009, 13440, ...
[5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 3, 2, 1;
[4] 0, 15, 8, 3, 1;
[5] 0, 105, 48, 15, 4, 1;
[6] 0, 945, 384, 105, 24, 5, 1;
.
From _Werner Schulte_, Mar 07 2024: (Start)
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 3 2 | * | 1 3 6 ... | = | 0 3 8 15 24 ... |
| 0 15 18 6 | | 1 4 ... | | 0 15 48 105 192 ... |
| 0 105 174 108 24 | | 1 ... | | 0 105 384 945 1920 ... |
| . . . | | . . . | | . . . |. (End)
Links
- Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
Crossrefs
Programs
-
Maple
A := (n, k) -> 2^n*pochhammer(k/2, 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 - 2*x)^(-k/2); ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end: seq(lprint(EGFcol(n, 9)), n = 0..8); # Using the generating polynomials for the rows: P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n): seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5); # Implementing the comment of Werner Schulte about the LU decomposition of A: with(LinearAlgebra): L := Matrix(7, 7, (n, k) -> A371025(n - 1, k - 1)): U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)): MatrixMatrixMultiply(L, Transpose(U)); # Peter Luschny, Mar 08 2024
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Mathematica
A370419[n_, k_] := 2^n*Pochhammer[k/2, n]; Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)
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SageMath
def A(n, k): return 2**n * rising_factorial(k/2, n) for n in range(6): print([A(n, k) for k in range(9)])
Formula
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-2)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
From Werner Schulte, Mar 07 2024: (Start)
A(n, k) = Product_{i=1..n} (2*i - 2 + k).
E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
E.g.f. of row n > 0: Sum_{k>=1} A(n, k) * x^k / (k!) = (Sum_{k=1..n} A035342(n, k) * x^k) * exp(x).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
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, where L is defined L(n, k) = A035342(n, k) * k! for 1 <= k <= n and L(n, 0) = 0^n. Note that L(n, k) + L(n, k+1) = A265649(n, k) * k! for 0 <= k <= n. (End)