A342889 Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_10 (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 11, 1, 1, 66, 66, 1, 1, 286, 1716, 286, 1, 1, 1001, 26026, 26026, 1001, 1, 1, 3003, 273273, 1184183, 273273, 3003, 1, 1, 8008, 2186184, 33157124, 33157124, 2186184, 8008, 1, 1, 19448, 14158144, 644195552, 2254684432, 644195552, 14158144, 19448, 1

Offset

0,5

References

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993.

Links

Seiichi Manyama, Rows n = 0..139, flattened

Richard L. Ollerton, Counting i-paths, Slides of talk presented at Thirteenth International Conference on Fibonacci Numbers and Their Applications, University of Patras (Greece), 2008.

Richard L. Ollerton, Counting i-paths, Background notes for slides of talk presented at Thirteenth International Conference on Fibonacci Numbers and Their Applications, University of Patras (Greece), 2008.

Richard L. Ollerton and Anthony G. Shannon, Extensions of generalized binomial coefficients, In: Howard F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht, pp. 187-199.

Richard L. Ollerton and Anthony G. Shannon, Further properties of generalized binomial coefficient k-extensions, Fibonacci Quarterly 43.2 (2005): 124-129.

Daniel B. Shapiro, Divisibility Properties of Integer Sequences, arXiv:2302.02243 [math.NT], 2023. See also Integers, (2023) Vol. 23, #A57.

Formula

The generalized binomial coefficient (n,k)_m = Product_{j=1..k} binomial(n+m-j,m)/binomial(j+m-1,m).

Example

Triangle begins:
  [1],
  [1, 1],
  [1, 11, 1],
  [1, 66, 66, 1],
  [1, 286, 1716, 286, 1],
  [1, 1001, 26026, 26026, 1001, 1],
  [1, 3003, 273273, 1184183, 273273, 3003, 1],
  [1, 8008, 2186184, 33157124, 33157124, 2186184, 8008, 1],
...

Maple

# Generalized binomial coefficient:
GBC := proc(n, k, m) local a, j;
a := mul((binomial(n+m-j, m)/binomial(j+m-1, m)), j=1..k);
end;
# Returns first M rows of triangle:
GBCT := proc(m, M) local a, b, n, k; global GBC;
a:=[];
for n from 0 to M do
  b:=[seq(GBC(n, k, m), k=0..n)];
  a:=[op(a), b];
od: a; end;
GBCT(10, 12);

Mathematica

f[n_, k_, m_] := Product[Binomial[n + m - j, m]/Binomial[j + m - 1, m], {j, k}]; Table[f[n, k, 10], {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 25 2023 *)

Prog

(PARI) f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
T(n, k) = f(n, k, 10); \\ Seiichi Manyama, Apr 02 2021

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: A157209 A014469 A157149 * A166979 A022174 A173006

Adjacent sequences: A342886 A342887 A342888 * A342890 A342891 A342892

Keyword

nonn,tabl

Author

N. J. A. Sloane, Apr 01 2021

Status

approved