A142467 -id:A142467 - cp's OEIS frontend

A342891 Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_12 (n >= 0, 0 <= k <= n).

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

N. J. A. Sloane, Apr 01 2021

For references, links, programs, etc., see earlier sequences in this series, especially A342889.

			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],
...

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

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

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

A005365 Hoggatt sequence with parameter d=7.

Original entry on oeis.org

1, 2, 10, 74, 782, 10562, 175826, 3457742, 78408332, 2005691690, 56970282514, 1772967273794, 59814500606018, 2168062920325850, 83802728579860658, 3432438439271783026, 148165335791410936770, 6708873999658599592672

Offset: 0

N. J. A. Sloane

nonn

Let V be the vector representation of SL(7) (of dimension 7) and let E be the exterior algebra of V (of dimension 128). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 03 2021

This is the number of 7-vicious walkers (aka vicious 7-watermelons) - see Essam and Guttmann (1995). This is the 7-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021

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..496
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

Row sums of A142467.

Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005366, A116925.

A142467:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..6]]) >;
A005365:= func< n | (&+[A142467(n,k): k in [0..n]]) >;
[A005365(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022

A005365[n_]:=HypergeometricPFQ[{-6-n,-5-n,-4-n,-3-n,-2-n,-1-n,-n},{2,3,4,5,6,7},-1] (* Richard L. Ollerton, Sep 13 2006 *)

a(n) = my(d=7); 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([-6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2,3,4,5,6,7], -1))
[A005365(n) for n in range(51)] # G. C. Greubel, Nov 13 2022

a(n) = Hypergeometric7F6([-6-n, -5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6, 7], -1). - Richard L. Ollerton, Sep 13 2006
a(n) = S(7,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016
a(n) ~ 6075 * 2^(7*n + 57) / (sqrt(7) * Pi^3 * n^24). - Vaclav Kotesovec, Apr 01 2021

More terms from Sean A. Irvine, May 29 2016

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

Seiichi Manyama, Apr 01 2021

Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)n where (n,k)_m is Product{j=1..k} binomial(n-j+m,m)/binomial(j-1+m,m).

			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;

Seiichi Manyama, Rows n = 0..50, flattened

Row sums gives A342967.

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);

T(n,k) = Product_{j=0..k-1} binomial(2*n-1,n+j)/binomial(2*n-1,j).

A142597 Triangle read by rows: t(n,k)=t(n - 1, k - 1) + 4* t(n - 1, k) + 3*t(n - 1, k - 1).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 36, 36, 1, 1, 148, 288, 148, 1, 1, 596, 1744, 1744, 596, 1, 1, 2388, 9360, 13952, 9360, 2388, 1, 1, 9556, 46992, 93248, 93248, 46992, 9556, 1, 1, 38228, 226192, 560960, 745984, 560960, 226192, 38228, 1, 1, 152916, 1057680, 3148608

Offset: 1

Roger L. Bagula, Sep 22 2008

Row sums are: {1, 2, 10, 74, 586, 4682, 37450, 299594, 2396746, 19173962, ...}.

			Triangle begins:
{1},
{1, 1},
{1, 8, 1},
{1, 36, 36, 1},
{1, 148, 288, 148, 1},
{1, 596, 1744, 1744, 596, 1},
{1, 2388, 9360, 13952, 9360, 2388, 1},
{1, 9556, 46992, 93248, 93248, 46992, 9556, 1},
{1, 38228, 226192, 560960, 745984, 560960, 226192, 38228, 1},
{1, 152916, 1057680, 3148608, 5227776, 5227776, 3148608, 1057680, 152916, 1}

Cf. A142175, A142467.

Cf. A008292, A119258, A060187, A142175, A142467.

A[n_, 1] := 1 A[n_, n_] := 1 A[n_, k_] := A[n - 1, k - 1] + 4* A[n - 1, k] + 3*A[n - 1, k - 1]; a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]

Edited by N. J. A. Sloane, Dec 07 2008

A155834 A triangle sequence of general recursive Sierpinski-Pascal minus general Narayana with adjusted n,m levels and zeros out:k=2; t(n,m)=Pascal(n,m,k-1)-Narayana(n-1,m-1,2*(k-1)).

Original entry on oeis.org

1, 1, 6, 16, 6, 22, 127, 127, 22, 64, 701, 1436, 701, 64, 163, 3117, 11503, 11503, 3117, 163, 382, 12088, 74122, 131494, 74122, 12088, 382, 848, 42890, 413612, 1193930, 1193930, 413612, 42890, 848, 1816, 143562, 2094588, 9280734, 14992440, 9280734

Offset: 4

Roger L. Bagula, Jan 28 2009

Row sums are;
2, 28, 298, 2966, 29566, 304678, 3302560, 38033840, 467861040, 6159690808,
86763791762,...
This level is the Eulerian number level:
only the odd Narayana levels correspond to the recursive Sierpinski-Pascal levels.

			{1, 1},
{6, 16, 6},
{22, 127, 127, 22},
{64, 701, 1436, 701, 64},
{163, 3117, 11503, 11503, 3117, 163},
{382, 12088, 74122, 131494, 74122, 12088, 382},
{848, 42890, 413612, 1193930, 1193930, 413612, 42890, 848},
{1816, 143562, 2094588, 9280734, 14992440, 9280734, 2094588, 143562, 1816},
{3797, 462541, 9928140, 64761204, 158774838, 158774838, 64761204, 9928140, 462541, 3797},
{7814, 1453700, 44960878, 418557816, 1489425900, 2250878592, 1489425900, 418557816, 44960878, 1453700, 7814},
{15914, 4495909, 197226603, 2558716162, 12781854516, 27839586777, 27839586777, 12781854516, 2558716162, 197226603, 4495909, 15914}

A001263, A056939.A056941, A142465, A142467

Clear[A, a0, b0, n, k, m, t, i];
A[n_, 1, m_] := 1; A[n_, n_, m_] := 1;
A[n_, k_, m_] := (m*n - m*k + 1)*A[n - 1, k - 1, m] + (m*k - (m - 1))*A[n - 1, k, m];
t[n_, m_, i_] = Product[Binomial[n + k, m + k]/Binomial[n - m + k, k], {k, 0, i}];
m = 2; a = Table[A[n, k, m - 1] - t[n - 1, k - 1, (2*m - 2)], {n, 4, 14}, { k, 2, n - 1}];
Flatten[a]

Pascal(n,m,k):
a(n,k,m)=(m*n - m*k + 1)*a(n - 1, k - 1, m) + (m*k - (m - 1))*a(n - 1, k, m);
Narayana(n,m,k):
y(n,m,k)=Product[Binomial[n + k, m + k]/Binomial[n - m + k, k], {k, 0, i}];
k=2;
t(n,m)=Pascal(n,m,k-1)-Narayana(n-1,m-1,2*(k-1)).

cp's OEIS Frontend

A342891 Triangle read by rows: T(n,k) = generalized binomial coefficients (n,k)_12 (n >= 0, 0 <= k <= n).

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A005365 Hoggatt sequence with parameter d=7.

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A342972 Triangle T(n,k) read by rows: T(n,k) = Product_{j=0..n-1} binomial(n+j,k)/binomial(k+j,k).

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A142597 Triangle read by rows: t(n,k)=t(n - 1, k - 1) + 4* t(n - 1, k) + 3*t(n - 1, k - 1).

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A155834 A triangle sequence of general recursive Sierpinski-Pascal minus general Narayana with adjusted n,m levels and zeros out:k=2; t(n,m)=Pascal(n,m,k-1)-Narayana(n-1,m-1,2*(k-1)).

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