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A342889
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Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_10 (n >= 0, 0 <= k <= n).
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14
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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
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table;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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The generalized binomial coefficient (n,k)_m = Product_{j=1..k} binomial(n+m-j,m)/binomial(j+m-1,m).
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EXAMPLE
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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],
...
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MAPLE
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# 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);
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MATHEMATICA
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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 *)
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PROG
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(PARI) f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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