A056940 -id:A056940 - 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).

A005362 Hoggatt sequence with parameter d=4.

Original entry on oeis.org

1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, 366039104, 3579512809, 36091415154, 373853631974, 3966563630394, 42997859838010, 475191259977060, 5344193918791710, 61066078557804360, 707984385321707910, 8318207051955884772, 98936727936728464152

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Let V be the vector representation of SL(4) (of dimension 4) and let E be the exterior algebra of V (of dimension 16). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 18 2021

This is the number of 4-vicious walkers (aka vicious 4-watermelons) - see Essam and Guttmann (1995). This is the 4-walker analog of A001181. - N. J. A. Sloane, Mar 22 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..845
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, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.
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)
Nick Hobson, Python program for this sequence
Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

Cf. A005364, A005365, A005366, A056940, A116925.

A056940:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..3]]) >;
A005362:= func< n | (&+[A056940(n,k): k in [0..n]]) >;
[A005362(n): n in [0..30]]; // G. C. Greubel, Nov 14 2022

a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021

A005362[n_]:=HypergeometricPFQ[{-3-n,-2-n,-1-n,-n},{2,3,4},1] (* Richard L. Ollerton, Sep 12 2006 *)

def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n],[2,3,4], 1))
[A005362(n) for n in range(41)] # G. C. Greubel, Nov 14 2022

From Richard L. Ollerton, Sep 12 2006: (Start)
a(n) = Hypergeometric4F3([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1).
(n+3)*(n+4)*(n+5)*(n+6)*a(n) = 6*(n+1)*(n+3)*(n+4)*(2*n+5)*a(n-1) + 4*(n-1)*n*(4*n+7)*(4*n+9)*a(n-2); a(0)=1, a(1)=2. (End)
a(n) = S(4,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016
a(n) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Apr 01 2021

A359363 Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 10, 1, 0, 1, 20, 50, 20, 1, 0, 1, 35, 175, 175, 35, 1, 0, 1, 56, 490, 980, 490, 56, 1, 0, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1, 0, 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1

Offset: 0

Peter Luschny, Dec 28 2022

This triangle is a member of a family of Pascal-like triangles. Let T(n, k, m) = sf(m)*F(n - 1) / (F(k - 1)*F(n - k)) if k > 0 and otherwise k^n, where F(n) = Product_{j=0..m} (n + j)! and sf(m) are the superfactorials A000178. The case m = 2 gives this triangle, some other cases are given in the crossreferences. See also A342889 for a related representation of generalized binomial coefficients.

			Triangle T(n, k) starts:
[0] 1
[1] 0, 1
[2] 0, 1,   1
[3] 0, 1,   4,    1
[4] 0, 1,  10,   10,     1
[5] 0, 1,  20,   50,    20,     1
[6] 0, 1,  35,  175,   175,    35,     1
[7] 0, 1,  56,  490,   980,   490,    56,    1
[8] 0, 1,  84, 1176,  4116,  4116,  1176,   84,   1
[9] 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1
.
Let p = (p1, p2,..., pn) denote a permutation of {1, 2,..., n}. The pair (p(i), p(i+1)) is a 'rise' if p(i) < p(i+1). Additionally a conventional rise is counted at the beginning of p.
T(n, k) is the number of Baxter permutations of {1,2,...,n} with k rises. For example for n = 4, [T(n, k) for k = 0..n] = [0, 1, 10, 10, 1]. The permutations, with preceding number of rises, are:
.
1 [4, 3, 2, 1],  3 [2, 3, 4, 1],  2 [3, 4, 2, 1],  3 [2, 3, 1, 4],
2 [3, 2, 4, 1],  3 [2, 1, 3, 4],  2 [3, 2, 1, 4],  3 [1, 3, 4, 2],
2 [2, 4, 3, 1],  3 [1, 3, 2, 4],  2 [4, 2, 3, 1],  3 [3, 4, 1, 2],
2 [2, 1, 4, 3],  3 [3, 1, 2, 4],  2 [4, 2, 1, 3],  3 [1, 2, 4, 3],
2 [1, 4, 3, 2],  3 [1, 4, 2, 3],  2 [4, 1, 3, 2],  3 [4, 1, 2, 3],
2 [4, 3, 1, 2],  4 [1, 2, 3, 4].

Special cases of the general formula: A097805 (m = 0), (0,1)-Pascal triangle; A090181 (m = 1), triangle of Narayana; this triangle (m = 2); A056940 (m = 3), with 1,0,0...; A056941 (m = 4), with 1,0,0...; A142465 (m = 5), with 1,0,0....
Variant: A056939. Diagonals: A000292, A006542, A047819.
Cf. A000178, A001181, A046996, A217800, A342889.

p := (n, x) -> ifelse(n = 0, 1, x*hypergeom([-1-n, -n, 1-n], [2, 3], -x)):
seq(seq(coeff(simplify(p(n, x)), x, k), k = 0..n), n = 0..10);
# Alternative:
T := proc(n, k) local F; F := n -> n!*(n+1)!*(n+2)!;
ifelse(k = 0, k^n, 2*F(n-1)/(F(k-1)*F(n-k))) end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

C=binomial;
T(n, k) = if(n==0 && k==0, 1, ( C(n+1,k-1) * C(n+1,k) * C(n+1,k+1) ) / ( C(n+1,1) * C(n+1,2) ) );
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print());
\\ Joerg Arndt, Jan 04 2024

from functools import cache
from math import factorial
@cache
def A359363Row(n: int) -> list[int]:
    @cache
    def F(n: int): return factorial(n) ** 3 * ((n+1) * (n+1) * (n+2))
    if n == 0: return [1]
    return [0] + [(2*F(n-1))//(F(k-1) * F(n-k)) for k in range(1, n+1)]
for n in range(0, 10): print(A359363Row(n))
# Peter Luschny, Jan 04 2024

def A359363(n):
    if n == 0: return SR(1)
    h = x*hypergeometric([-1 - n, -n, 1 - n], [2, 3], -x)
    return h.series(x, n + 1).polynomial(SR)
for n in range(10): print(A359363(n).list())
def PolyA359363(n, t): return Integer(A359363(n)(x=t).n())
# Peter Luschny, Jan 04 2024

T(n, k) = [x^k] p(n, x).
T(n, k) = 2*F(n-1)/(F(k-1)*F(n-k)) for k > 0 where F(n) = n!*(n+1)!*(n+2)!.
p(n, 1) = A001181(n), i.e. the Baxter numbers are the values of the Baxter polynomials at x = 1.
(-1)^(n + 1)*p(2*n + 1, -1) = A217800(n) .

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 20, 4, 1, 1, 70, 175, 50, 5, 1, 1, 252, 1764, 980, 105, 6, 1, 1, 924, 19404, 24696, 4116, 196, 7, 1, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1

Offset: 0

Paul Barry, Jun 13 2006

Columns are the central coefficients of the triangles T(n, k;r) with T(n, k;r)=Product{j=0..r, C(n+j, k+j)/C(n-k+j, j)}*[k<=n]; (r=0,A007318), (r=1;A001263),(r=2,A056939),(r=3,A056940),(r=4,A056941). Essentially A103905 as a number triangle with an extra diagonal of 1's. Central coefficients T(2n, n) are A008793. Row sums are A120259. Diagonal sums are A120260.

			Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   6,     3,     1;
  1,  20,    20,     4,    1;
  1,  70,   175,    50,    5,   1;
  1, 252,  1764,   980,  105,   6, 1;
  1, 924, 19404, 24696, 4116, 196, 7, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A120259.

Cf. A000891, A000984, A103905, A120257.

T(n, k) = prod(j=0, k-1, binomial(2*n-2*k+j, n-k)/binomial(n-k+j, j)); \\ Seiichi Manyama, Apr 02 2021

Number triangle T(n, k)=[k<=n]*Product{j=0..k-1, C(2n-2k+j, n-k)/C(n-k+j, j)}

As a square array, this is T(n,m)=product{k=1..m, product{j=1..n, product{i=1..n, (i+j+k-1)/(i+j+k-2)}}}; - Paul Barry, May 13 2008

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).

A120257 Triangle of Hankel transforms of certain binomial sums.

Original entry on oeis.org

1, 2, -1, 3, -6, -1, 4, -20, -20, 1, 5, -50, -175, 70, 1, 6, -105, -980, 1764, 252, -1, 7, -196, -4116, 24696, 19404, -924, -1, 8, -336, -14112, 232848, 731808, -226512, -3432, 1, 9, -540, -41580, 1646568, 16818516, -24293412, -2760615, 12870, 1, 10, -825, -108900, 9343620, 267227532, -1447482465

Offset: 0

Paul Barry, Jun 13 2006

Row k is the Hankel transform of Sum_{j=0..n} binomial(k+j, j). Absolute value is reversal of A103905. Diagonal and subdiagonals are essentially signed versions of the central coefficients of certain generalized Pascal-Narayana triangles (A007318, A001263, A056939, A056940, A056941).

			Triangle begins
  1;
  2,   -1;
  3,   -6,     -1;
  4,  -20,    -20,      1;
  5,  -50,   -175,     70,      1;
  6, -105,   -980,   1764,    252,      -1;
  7, -196,  -4116,  24696,  19404,    -924,    -1;
  8, -336, -14112, 232848, 731808, -226512, -3432, 1;

Cf. A120258.

T(n, k) = (-1)^((k+1)\2) * prod(j=0, n-k-1, binomial(2*k+2+j, k+1)/binomial(k+1+j, j)); \\ Michel Marcus, Jan 13 2022

T(n, k) = (cos(Pi*k/2) - sin(Pi*k/2)) * Product_{j=0..n-k-1} C(2k+2+j, k+1)/C(k+1+j, j).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A005362 Hoggatt sequence with parameter d=4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A359363 Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Python

SageMath

Formula

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A120257 Triangle of Hankel transforms of certain binomial sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula