OFFSET
0,5
COMMENTS
Triangle of generalized binomial coefficients (n,k)_5; cf. A342889. - N. J. A. Sloane, Apr 03 2021
REFERENCES
Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
P. A. MacMahon, Combinatory Analysis, Section 495, 1916.
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
LINKS
Seiichi Manyama, Rows n = 0..139 of triangle, flattened
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
P. A. MacMahon, Combinatory analysis.
FORMULA
From Peter Bala, Oct 13 2011: (Start)
A(n, k) = Product_{j=0..4} C(n+k+j, k+j)/C(n+j, j) gives the array as a square.
g(n-1, k-1)*g(n, k+1)*g(n+1, k) = g(n-1, k)*g(n, k-1)*g(n+1, k+1) where g(n, k) is the array A(n, k) and triangle T(n, k).
Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)
From Peter Bala, May 10 2012: (Start)
Determinants of 5 X 5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present).
Also determinants of 5 X 5 arrays whose entries come from a single row:
det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3), C(n,k-4); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+4), C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. (End)
From G. C. Greubel, Nov 14 2022: (Start)
T(n, k) = Product_{j=0..4} binomial(n+j, k)/binomial(k+j, k) (gives the triangle).
Sum_{k=0..n} T(n, k) = A005363(n). (End)
EXAMPLE
The array starts:
[1 1 1 1 1 1 1 ...]
[1 6 21 56 126 252 462 ...]
[1 21 196 1176 5292 19404 60984 ...]
[1 56 1176 14112 116424 731808 3737448 ...]
[1 126 5292 116424 1646568 16818516 133613766 ...]
[1 252 19404 731808 16818516 267227532 3184461423 ...]
[1 462 60984 3737448 133613766 3184461423 55197331332 ...]
[...]
Considered as a triangle, the initial rows are:
1;
1, 1;
1, 6, 1;
1, 21, 21, 1;
1, 56, 196, 56, 1;
1, 126, 1176, 1176, 126, 1;
1, 252, 5292, 14112, 5292, 252, 1;
1, 462, 19404, 116424, 116424, 19404, 462, 1;
1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1; ...
MATHEMATICA
T[n_, k_] := Product[Binomial[n+j, k]/Binomial[k+j, k], {j, 0, 4}];
Table[T[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 14 2022 *)
PROG
(PARI) A056941(n, m)=prod(k=0, 4, binomial(n+m+k, m+k)/binomial(n+k, k)) \\ as an array \\ M. F. Hasler, Sep 26 2018
(Magma)
A056941:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..4]]) >;
[A056941(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 14 2022
(SageMath)
def A056941(n, k): return product(binomial(n+j, k)/binomial(k+j, k) for j in (0..4))
flatten([[A056941(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 14 2022
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
Edited by M. F. Hasler, Sep 26 2018
STATUS
approved