A005366 -id:A005366 - cp's OEIS frontend

A005362 Hoggatt sequence with parameter d=4.

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

A005363 Hoggatt sequence with parameter d=5.

Original entry on oeis.org

1, 2, 8, 44, 310, 2606, 25202, 272582, 3233738, 41454272, 567709144, 8230728508, 125413517530, 1996446632130, 33039704641922, 566087847780250, 10006446665899330, 181938461947322284, 3393890553702212368, 64807885247524512668, 1264344439859632559216

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

Let V be the vector representation of SL(5) (of dimension 5) and let E be the exterior algebra of V (of dimension 32). 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 5-vicious walkers (aka vicious 5-watermelons) - see Essam and Guttmann (1995). This is the 5-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..681
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)
Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

Cf. A000079, A000108, A001181, A005362, A005364, A005365, A005366, A116925.

Cf. A056941.

A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;
A005363:= func< n | (&+[A056941(n,k): k in [0..n]]) >;
[A005363(n): n in [0..40]]; // G. C. Greubel, Nov 14 2022

a := n -> hypergeom([-4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5], -1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021
# The following Maple program is based on Eq (60) of Essam-Guttmann (1995) and confirms that that sequence is the same as the present one. - N. J. A. Sloane, Mar 27 2021
v5 := proc(n) local t1,t2,t3,t4,t5;
if n=0 then 1
elif n=1 then 2
elif n=2 then 8
else
t1 := (4+n)*(5+n)^2*(6+n)*(7+n)*(8+n)*(252+253*n+55*n^2);
t2 := 3*(4+n)*(5+n)*(141120+362152*n + 373054*n^2+192647*n^3+52441*n^4 +7161*n^5 +385*n^6);
t3 := n*(1-n)*(5738880+14311976*n+14466242*n^2+7579175*n^3 +2170343*n^4+322289*n^5 + 19415*n^6);
t4 := 32*(2-n)*(1-n)^2*n^2*(1+n)*(560+363*n+55*n^2);
t5 := t2*v5(n-1)-t3*v5(n-2)+t4*v5(n-3);
t5/t1;
fi; end;
[seq(v5(n), n=0..20)];

A005363[n_]:=HypergeometricPFQ[{-4-n,-3-n,-2-n,-1-n,-n},{2,3,4,5},-1] (* Richard L. Ollerton, Sep 12 2006 *)

def A005363(n): return simplify(hypergeometric([-4-n, -3-n, -2-n, -1-n, -n],[2,3,4,5], -1))
[A005363(n) for n in range(51)] # G. C. Greubel, Nov 14 2022

From Richard L. Ollerton, Sep 12 2006: (Start)
a(n) = Hypergeometric5F4([-4-n, -3-n, -2-n, -1-n, -n], [2,3,4,5], -1).
(n+4)*(n+5)^2*(n+6)*(n+7)*(n+8)*(252 +253*n +55*n^2)*a(n) = 3*(n+4)*(n+5)*(141120 + 362152*n + 373054*n^2 + 192647*n^3 + 52441*n^4 + 7161*n^5 + 385*n^6)*a(n-1) + n*(n-1)*(5738880 + 14311976*n + 14466242*n^2 + 7579175*n^3 + 2170343*n^4 + 322289*n^5 + 19415*n^6)*a(n-2) - 32*(n-1)^2*n^2*(n-2)*(n+1)*(560 + 363*n + 55*n^2)*a(n-3); a(-1)=a(0)=1, a(1)=2. (End)
a(n) = S(5,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016
a(n) ~ 9 * 2^(5*n + 27) / (sqrt(5) * Pi^2 * n^12). - Vaclav Kotesovec, Apr 01 2021
a(n) = Sum_{k=0..n} A056941(n, k) (row sums of triangle A056941). - G. C. Greubel, Nov 14 2022

More terms from Sean A. Irvine, May 29 2016

A005364 Hoggatt sequence with parameter d=6.

Original entry on oeis.org

1, 2, 9, 58, 506, 5462, 70226, 1038578, 17274974, 317292692, 6346909285, 136723993122, 3143278648954, 76547029418394, 1962350550273130, 52679691605422354, 1474290522744355250, 42847373913958703100, 1288899422418558314550, 40013380588722843337620

Offset: 0

N. J. A. Sloane

nonn

Let V be the vector representation of SL(6) (of dimension 6) and let E be the exterior algebra of V (of dimension 64). 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 6-vicious walkers (aka vicious 6-watermelons) - see Essam and Guttmann (1995). This is the 6-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..573
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 A142465.

Cf. A000079, A000108, A001181, A005362, A005363, A005365, A005366, A116925.

A142465:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..5]]) >;
A005364:= func< n | (&+[A142465(n,k): k in [0..n]]) >;
[A005364(n): n in [0..40]]; // G. C. Greubel, Nov 13 2022

A005364[n_]:=HypergeometricPFQ[{-5-n,-4-n,-3-n,-2-n,-1-n,-n},{2,3,4,5,6},1] (* Richard L. Ollerton, Sep 13 2006 *)

a(n) = my(d=6); 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 A005364(n): return simplify(hypergeometric([-5-n, -4-n, -3-n, -2-n, -1-n, -n],[2, 3, 4, 5, 6], 1))
[A005364(n) for n in range(51)] # G. C. Greubel, Nov 13 2022

a(n) = Hypergeometric6F5([-5-n, -4-n, -3-n, -2-n, -1-n, -n], [2, 3, 4, 5, 6], 1). - Richard L. Ollerton, Sep 13 2006
a(n) = S(6,n) where S(d,n) = 1 + Sum_{h=0..n-1} Product_{k=0..h} binomial(n+d-1-k,d) / binomial(d + k, d) [From Fielder and Alford]. - Sean A. Irvine, May 29 2016
a(n) ~ 135 * 2^(6*n + 40) / (sqrt(3) * Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Apr 01 2021

More terms from Sean A. Irvine, May 29 2016

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

A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2n-1,n+k) / binomial(2n-1,k).

Original entry on oeis.org

1, 2, 5, 22, 177, 2606, 70226, 3457742, 311348897, 51177188350, 15377065068510, 8430169458379450, 8446194335222422950, 15435904380166258833482, 51546769958534244310727102, 313937270864810066000897492222, 3493348088919874482660174997662017

Offset: 0

Seiichi Manyama, Apr 01 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..87

Row sums of A342972.

Cf. A001181, A005362, A005363, A005364, A005365, A005366, A116925.

a[n_] := 1 + Sum[Product[Binomial[2*n - 1, n + k]/Binomial[2*n - 1, k], {k, 0, j - 1}], {j, 1, n}]; Array[a, 17, 0] (* Amiram Eldar, Apr 01 2021 *)
Table[1 + BarnesG[2*n + 1] * Sum[BarnesG[j + 1]*BarnesG[n - j + 1] / (BarnesG[n + j + 1]*BarnesG[2*n - j + 1]), {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 02 2021 *)

a(n) = 1+sum(j=1, n, prod(k=0, j-1, binomial(2*n-1, n+k)/binomial(2*n-1, k)));

a(n) = sum(j=0, n, prod(k=0, n-1, binomial(n+k, j)/binomial(j+k, j)));

a(n) = Sum_{j=0..n} Product_{k=0..n-1} binomial(n+k,j)/binomial(j+k,j).

a(n) ~ c * exp(1/12) * 2^(4*n^2 - 1/12) / (A * n^(1/12) * 3^(9*n^2/4 - 1/6)), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd, and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 02 2021

cp's OEIS Frontend

A005362 Hoggatt sequence with parameter d=4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A005363 Hoggatt sequence with parameter d=5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A005364 Hoggatt sequence with parameter d=6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A005365 Hoggatt sequence with parameter d=7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2*n-1,n+k) / binomial(2*n-1,k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A342967 a(n) = 1 + Sum_{j=1..n} Product_{k=0..j-1} binomial(2n-1,n+k) / binomial(2n-1,k).