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A142465 Triangle T(n,m) read by rows: T(n,m) = Product_{i=0..5} binomial(n+i,m)/binomial(m+i,m). 16
1, 1, 1, 1, 7, 1, 1, 28, 28, 1, 1, 84, 336, 84, 1, 1, 210, 2520, 2520, 210, 1, 1, 462, 13860, 41580, 13860, 462, 1, 1, 924, 60984, 457380, 457380, 60984, 924, 1, 1, 1716, 226512, 3737448, 9343620, 3737448, 226512, 1716, 1, 1, 3003, 736164, 24293412, 133613766, 133613766, 24293412, 736164, 3003, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Triangle of generalized binomial coefficients (n,k)_6; cf. A342889. - N. J. A. Sloane, Apr 03 2021
The matrix inverse starts
1;
-1, 1;
6, -7, 1
-141, 168, -28, 1;
9911, -11844, 2016, -84, 1;
-1740901, 2081310, -355320, 15120, -210, 1. - R. J. Mathar, Mar 22 2013
LINKS
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).
FORMULA
T(n,m) = A056941(n,m)*binomial(n+5,m)/binomial(m+5,m).
Sum_{k=0..n} T(n, k) = A005364(n).
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 7, 1;
1, 28, 28, 1;
1, 84, 336, 84, 1;
1, 210, 2520, 2520, 210, 1;
1, 462, 13860, 41580, 13860, 462, 1;
1, 924, 60984, 457380, 457380, 60984, 924, 1;
1, 1716, 226512, 3737448, 9343620, 3737448, 226512, 1716, 1;
1, 3003, 736164, 24293412, 133613766, 133613766, 24293412, 736164, 3003, 1;
MAPLE
A142465 := proc(n, m)
mul(binomial(n+i, m)/binomial(m+i, m), i=0..5) ;
end proc; # R. J. Mathar, Mar 22 2013
MATHEMATICA
T[n_, k_]:= Product[Binomial[n+j, k]/Binomial[k+j, k], {j, 0, 5}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = prod(j=0, 5, binomial(n+j, k)/binomial(k+j, k)); \\ Seiichi Manyama, Apr 01 2021
(Magma)
A142465:= func< n, k | (&*[Binomial(n+j, k)/Binomial(k+j, k): j in [0..5]]) >;
[A142465(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 13 2022
(SageMath)
def A142465(n, k): return product(binomial(n+j, k)/binomial(k+j, k) for j in (0..5))
flatten([[A142465(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 13 2022
CROSSREFS
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
Sequence in context: A179837 A238743 A168517 * A248829 A154337 A033933
KEYWORD
nonn,easy,tabl
AUTHOR
Roger L. Bagula, Sep 20 2008, Jan 28 2009
EXTENSIONS
Edited by the Associate Editors of the OEIS, May 17 2009
STATUS
approved

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Last modified March 28 14:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)