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%I #45 Mar 20 2024 17:01:54
%S 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,
%T 0,10395,3840,945,192,35,6,1,0,135135,46080,10395,1920,315,48,7,1,0,
%U 2027025,645120,135135,23040,3465,480,63,8,1
%N A(n, k) = 2^n*Pochhammer(k/2, n). Square array read by ascending antidiagonals.
%H Paolo Xausa, <a href="/A370419/b370419.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%F 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).
%F From _Werner Schulte_, Mar 06 and 07 2024: (Start)
%F A(n, k) = Product_{i=1..n} (2*i - 2 + k).
%F E.g.f. of column k: Sum_{n>=0} A(n, k) * t^n / (n!) = (1/sqrt(1 - 2*t))^k.
%F A(n, k) = A(n+1, k-2) / (k - 2) for k > 2.
%F A(n, k) = Sum_{i=0..k-1} i! * A265649(n, i) * binomial(k-1, i) for k > 0.
%F 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).
%F Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (k! * n!) = exp(x/sqrt(1 - 2*t)).
%F Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1 / (1 - x/sqrt(1 - 2*t)).
%F 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)
%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, 3, 8, 15, 24, 35, 48, 63, 80, ...
%e [3] 0, 15, 48, 105, 192, 315, 480, 693, 960, ...
%e [4] 0, 105, 384, 945, 1920, 3465, 5760, 9009, 13440, ...
%e [5] 0, 945, 3840, 10395, 23040, 45045, 80640, 135135, 215040, ...
%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, 3, 2, 1;
%e [4] 0, 15, 8, 3, 1;
%e [5] 0, 105, 48, 15, 4, 1;
%e [6] 0, 945, 384, 105, 24, 5, 1;
%e .
%e From _Werner Schulte_, Mar 07 2024: (Start)
%e Illustrating the LU decomposition of A:
%e / 1 \ / 1 1 1 1 1 ... \ / 1 1 1 1 1 ... \
%e | 0 1 | | 1 2 3 4 ... | | 0 1 2 3 4 ... |
%e | 0 3 2 | * | 1 3 6 ... | = | 0 3 8 15 24 ... |
%e | 0 15 18 6 | | 1 4 ... | | 0 15 48 105 192 ... |
%e | 0 105 174 108 24 | | 1 ... | | 0 105 384 945 1920 ... |
%e | . . . | | . . . | | . . . |. (End)
%p A := (n, k) -> 2^n*pochhammer(k/2, 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 - 2*x)^(-k/2);
%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..8);
%p # Using the generating polynomials for the rows:
%p P := (n, x) -> local k; add(Stirling1(n, k)*(-2)^(n - k)*x^k, k=0..n):
%p seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
%p # Implementing the comment of _Werner Schulte_ about the LU decomposition of A:
%p with(LinearAlgebra):
%p L := Matrix(7, 7, (n, k) -> A371025(n - 1, k - 1)):
%p U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
%p MatrixMatrixMultiply(L, Transpose(U)); # _Peter Luschny_, Mar 08 2024
%t A370419[n_, k_] := 2^n*Pochhammer[k/2, n];
%t Table[A370419[n-k, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Mar 06 2024 *)
%o (SageMath)
%o def A(n, k): return 2**n * rising_factorial(k/2, n)
%o for n in range(6): print([A(n, k) for k in range(9)])
%Y Columns: A000007, A001147, A000165, A001147 (shifted), A002866, A051577, A051578, A051579, A051580.
%Y Rows: A000012, A001477, A005563, A370912, A190577.
%Y Cf. A035342, A265649, A370890, A370982 (row sums of the triangle), A370915, A371025, A371077.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Mar 04 2024
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