A371027
a(n) = Sum_{k=1..n} binomial(n, k) * Pochhammer(1/4, k) * 4^k * a(n - k) for n > 0 and a(0) = 1. Row sums of triangle A371026.
Original entry on oeis.org
1, 1, 7, 81, 1299, 26565, 659655, 19256265, 645765435, 24453391725, 1031685317775, 47978413923825, 2437962646702275, 134374287343821525, 7984095179348863575, 508687013361879977625, 34593307096333435324875, 2500919993138258679112125, 191526329081848509391017375
Offset: 0
-
a := proc(n) option remember; local k; if n = 0 then 1 else
add(binomial(n, k) * pochhammer(1/4, k) * 4^k * a(n - k), k = 1..n) fi end:
seq(a(n), n = 0..19);
A370915
A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.
Original entry on oeis.org
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
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;
Columns:
A000007,
A007696,
A001813,
A008545,
A047053,
A007696,
A000407,
A034176,
A052570 and
A034177,
A051617,
A051618,
A051619,
A051620.
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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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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 *)
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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)])
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