A052570 -id:A052570 - cp's OEIS frontend

A034177 a(n) is the n-th quartic factorial number divided by 4.

1, 8, 96, 1536, 30720, 737280, 20643840, 660602880, 23781703680, 951268147200, 41855798476800, 2009078326886400, 104472072998092800, 5850436087893196800, 351026165273591808000, 22465674577509875712000, 1527665871270671548416000, 109991942731488351485952000

Offset: 1

Wolfdieter Lang

			G.f. = x + 8*x^2 + 96*x^3 + 1536*x^4 + 30720*x^5 + 737820*x^6 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..365
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 513.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A007696, A000407, A034176. First column of triangle A048786.
A052570 is an essentially identical sequence. - Philippe Deléham, Sep 18 2008
Equals the second right hand column of A167569 divided by 2. - Johannes W. Meijer, Nov 12 2009

List([1..20], n-> 4^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 15 2019

[4^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019

[seq(n!*4^(n-1), n=1..16)]; # Zerinvary Lajos, Sep 23 2006

Array[4^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 4^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019

[4^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

4*a(n) = (4*n)(!^4) = Product_{j=1..n} 4*j = 4^n * n!.
E.g.f.: (-1 + 1/(1-4*x))/4.
D-finite with recurrence: a(n) -4*n*a(n-1)=0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(exp(1/4)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(1-exp(-1/4)). (End)

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

Peter Luschny, Mar 06 2024

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

			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;

Similar square arrays: A094587, A370419, A371077.
Rows: A000012, A001477, A028347, A370914.
Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.
Cf. A370913 (row sums of triangle), A371026.

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));

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 *)

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)])

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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A034177 a(n) is the n-th quartic factorial number divided by 4.

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