A342891
Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_12 (n >= 0, 0 <= k <= n).
Original entry on oeis.org
1, 1, 1, 1, 13, 1, 1, 91, 91, 1, 1, 455, 3185, 455, 1, 1, 1820, 63700, 63700, 1820, 1, 1, 6188, 866320, 4331600, 866320, 6188, 1, 1, 18564, 8836464, 176729280, 176729280, 8836464, 18564, 1, 1, 50388, 71954064, 4892876352, 19571505408, 4892876352, 71954064, 50388, 1
Offset: 0
Triangle begins:
[1],
[1, 1],
[1, 13, 1],
[1, 91, 91, 1],
[1, 455, 3185, 455, 1],
[1, 1820, 63700, 63700, 1820, 1],
[1, 6188, 866320, 4331600, 866320, 6188, 1],
[1, 18564, 8836464, 176729280, 176729280, 8836464, 18564, 1],
...
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.
-
f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
T(n, k) = f(n, k, 12); \\ Seiichi Manyama, Apr 02 2021
A005366
Hoggatt sequence with parameter d=8.
Original entry on oeis.org
1, 2, 11, 92, 1157, 19142, 403691, 10312304, 311348897, 10826298914, 426196716090, 18700516849302, 903666922873158, 47592378143008974, 2708388575679431454, 165309083872549538190, 10753269337589887334670, 741379205762167719365268
Offset: 0
- D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt sums and Hoggatt triangles, in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Seiichi Manyama, Table of n, a(n) for n = 0..438
- J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).
- D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988
- D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
- Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366
-
A142468:= func< n,k | Binomial(n,k)*(&*[Binomial(n+2*j,k+j)/Binomial(n+2*j,j): j in [1..7]]) >;
A005366:= func< n | (&+[A142468(n,k): k in [0..n]]) >;
[A005366(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022
-
A005366[n_]:=HypergeometricPFQ[{-7-n,-6-n,-5-n,-4-n,-3-n,-2-n,-1-n,-n},{2,3,4,5,6,7,8},1] (* Richard L. Ollerton, Sep 13 2006 *)
-
a(n) = my(d=8); 1 + sum(h=0, n-1, prod(k=0, h, binomial(n+d-1-k,d) / binomial(d + k, d))); \\ Michel Marcus, Feb 08 2021
-
def A005365(n): return simplify(hypergeometric([-7-n, -6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n],[2, 3, 4, 5, 6, 7, 8], 1))
[A005365(n) for n in range(51)] # G. C. Greubel, Nov 13 2022
A342972
Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 35, 105, 35, 1, 1, 126, 1176, 1176, 126, 1, 1, 462, 13860, 41580, 13860, 462, 1, 1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1, 1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1
Offset: 0
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 10, 10, 1;
1, 35, 105, 35, 1;
1, 126, 1176, 1176, 126, 1;
1, 462, 13860, 41580, 13860, 462, 1;
1, 1716, 169884, 1557270, 1557270, 169884, 1716, 1;
1, 6435, 2147145, 61408347, 184225041, 61408347, 2147145, 6435, 1;
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.
-
T[n_, k_] := Product[Binomial[n + i, k]/Binomial[k + i, k], {i, 0, n - 1}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 01 2021 *)
-
T(n, k) = prod(j=0, n-1, binomial(n+j, k)/binomial(k+j, k));
-
T(n, k) = prod(j=0, k-1, binomial(2*n-1, n+j)/binomial(2*n-1, j));
-
f(n, k, m) = prod(j=1, k, binomial(n-j+m, m)/binomial(j-1+m, m));
T(n, k) = f(n, k, n);
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