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

Author

N. J. A. Sloane, Apr 01 2021

Keywords

• nonn
• tabl

Examples

			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],
...
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Seiichi Manyama, Rows n = 0..139, flattened
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993.
• 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.

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.

Programs

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 *)

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

Formula

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