A005362 -id:A005362 - cp's OEIS frontend

A056940 Number of antichains (or order ideals) in the poset 4mn or plane partitions with at most m rows and n columns and entries <= 4.

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 35, 105, 35, 1, 1, 70, 490, 490, 70, 1, 1, 126, 1764, 4116, 1764, 126, 1, 1, 210, 5292, 24696, 24696, 5292, 210, 1, 1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1, 1, 495, 32670, 457380, 1646568, 1646568, 457380, 32670, 495, 1

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">4; cf. A342889. - _N. J. A. Sloane, Apr 03 2021
Determinants of 4 X 4 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005
Row sums are: {1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, ...}. - Roger L. Bagula, Mar 08 2010
Also determinants of 4x4 arrays whose entries come from a single row: T(n,k) = det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. - Peter Bala, May 10 2012

			Triangle begins as:
  1.
  1,   1.
  1,   5,     1.
  1,  15,    15,      1.
  1,  35,   105,     35,      1.
  1,  70,   490,    490,     70,      1.
  1, 126,  1764,   4116,   1764,    126,     1.
  1, 210,  5292,  24696,  24696,   5292,   210,   1.
  1, 330, 13860, 116424, 232848, 116424, 13860, 330, 1. - _Roger L. Bagula_, Mar 08 2010

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
P. A. MacMahon, Combinatory analysis, sect. 495, 1916.
R. P. Stanley, Theory and application of plane partitions, II. Studies in Appl. Math. 50 (1971), p. 259-279. DOI:10.1002/sapm1971503259. Thm. 18.1.
Index entries for sequences related to posets

Cf. A000372, A056932, A001263.
Antidiagonals sum to A005362 (Hoggatt sequence).
Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), A174109 (q=8).
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.

c[n_, q_] = Product[i + j, {j, 0, q}, {i, 1, n}];
T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Mar 08 2010 *)(* modified by G. C. Greubel, Apr 13 2019 *)

A056940(n,m)=prod(k=0,3,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ M. F. Hasler, Sep 26 2018

Product_{k=0..3} C(n+m+k, m+k)/C(n+k, k) gives the array as a square.
T(n,m,q) = c(n,q)/(c(m,q)*c(n-m,q)) with c(n,q) = Product_{i=1..n, j=0..q} (i + j), q = 3. - Roger L. Bagula, Mar 08 2010
From Peter Bala, Oct 13 2011: (Start)
T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).
Define f(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is f(r,0)*f(r,n)/(f(r,k)*f(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)

Edited by M. F. Hasler, Sep 26 2018

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

A116925 Triangle read by rows: row n (n >= 0) consists of the elements g(i, n-i) (0 <= i <= n), where g(r,s) = 1 + Sum_{k=1..r} Product_{i=0..k-1} binomial(r+s-1, s+i) / binomial(r+s-1, i).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 5, 8, 5, 1, 2, 6, 14, 16, 6, 1, 2, 7, 22, 42, 32, 7, 1, 2, 8, 32, 92, 132, 64, 8, 1, 2, 9, 44, 177, 422, 429, 128, 9, 1, 2, 10, 58, 310, 1122, 2074, 1430, 256, 10, 1, 2, 11, 74, 506, 2606, 7898, 10754, 4862, 512, 11, 1, 2, 12, 92, 782, 5462, 25202, 60398, 58202, 16796, 1024, 12

Offset: 0

Gary W. Adamson, Feb 26 2006

A generalized Catalan number triangle.
An alternative construction of this triangle. Begin with the Pascal triangle array, written as:
  1   1   1   1   1   1 ...
  1   2   3   4   5   6 ...
  1   3   6  10  15  21 ...
  1   4  10  20  35  56 ...
  1   5  15  35  70 126 ...
  ...
For each row r (r >= 0) in the above array, construct a triangle U(r) by applying the operation H defined below.
Then the r-th diagonal from the right in the new triangle is given by the row sums of U(r).
To define H, let us use row r=2, {1 3 6 10 15 ...}, as an illustration.
To get the 4th entry, take the first 4 terms of the row, reverse them and write them under the first 4 terms:
A:  1 3 6 10
B: 10 6 3  1
and form a new row C by beginning with 1 and iterating the map C' = C*B/A until we reach 1:
C: 1 10 20 10 1
E.g., 20 = (6 *10) / 3.
The sum of the terms {1 10 20 10 1} is 42, which is the 4th entry in the r=2 diagonal of the new triangle.
The full triangle U(2) begins
  1
  1  1
  1  3  1
  1  6  6  1
  1 10 20 10 1
...
(this is the Narayana triangle A001263)
and the row sums are the Catalan numbers, which give our r=2 diagonal.

			The first few rows of the triangle are:
  1
  1 2
  1 2  3
  1 2  4  4
  1 2  5  8   5
  1 2  6 14  16    6
  1 2  7 22  42   32    7
  1 2  8 32  92  132   64    8
  1 2  9 44 177  422  429  128   9
  1 2 10 58 310 1122 2074 1430 256 10
  ...

N. J. A. Sloane, First 30 rows, flattened

Diagonals of the triangle are generalized Catalan numbers. The first few diagonals (from the right) are A000027, A000079, A000108, A001181, A005362, A005363, ... The intermediate triangles include Pascal's triangle A007318, the Narayana triangle A001263, ...

Row sums give A104253.

g:=proc(n,p) local k,i; 1 + add( mul( binomial(n+p-1,p+i) / binomial(n+p-1,i), i=0..k-1 ), k=1..n); end; (N. J. A. Sloane, based on the formula from Hsueh-Hsing Hung)
f:=proc(n,r) local k,b,i; b:=binomial; add( mul( b(n+r-2,k-1+i),i=0..r-1)/ mul( b(n+r-2,i),i=1..r-1),k=1..n); end; M:=30; for j from 0 to M do lprint(seq(f(i,j+1-i),i=1..j+1)); od; # N. J. A. Sloane

rows = 11; t[n_, p_] := 1 + Sum[Product[ Binomial[ n+p-1, p+i] / Binomial[ n+p-1, i], {i, 0, k-1}], {k, 1, n}]; Flatten[ Table[ t[p, n-p], {n, 0, rows}, {p, 0, n}]](* Jean-François Alcover, Nov 18 2011, after Maple *)

Comment from N. J. A. Sloane, Sep 07 2006: (Start)
The n-th entry in the r-th diagonal from the right (r >= 0, n >= 1) is given by the quotient:
   Sum_{k=1..n} Product_{i=0..r-1} binomial(n+r-2, k-1+i)
   ------------------------------------------------------
           Product_{i=1..r-1} binomial(n+r-2, i)
(End)

One entry corrected by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 06 2006
Edited and extended by N. J. A. Sloane, Sep 07 2006
Simpler formula provided by Hsueh-Hsing Hung (hhh(AT)mail.nhcue.edu.tw), Sep 08 2006, which is now taken as the definition of this triangle
Edited by Jon E. Schoenfield, Dec 12 2015

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

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

N. J. A. Sloane

nonn

Let V be the vector representation of SL(8) (of dimension 8) and let E be the exterior algebra of V (of dimension 256). 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 8-vicious walkers (aka vicious 8-watermelons) - see Essam and Guttmann (1995). This is the 8-walker analog of A001181. - N. J. A. Sloane, Mar 27 2021
In general, for d > 0, a(n) ~ BarnesG(d+1) * 2^(d*n + (2*d+1)*(d-1)/2) / (sqrt(d) * Pi^((d-1)/2) * n^((d^2 - 1)/2)). - Vaclav Kotesovec, Apr 01 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..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

Row sums of A142468.

Cf. A000079, A000108, A001181, A005362, A005363, A005364, A005365, A116925.

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

a(n) = Hypergeometric8F7([-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) = S(8,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016
a(n) ~ 1913625 * 2^(8*n + 74) / (Pi^(7/2) * n^(63/2)). - 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

A056940 Number of antichains (or order ideals) in the poset 4*m*n or plane partitions with at most m rows and n columns and entries <= 4.

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A005363 Hoggatt sequence with parameter d=5.

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A116925 Triangle read by rows: row n (n >= 0) consists of the elements g(i, n-i) (0 <= i <= n), where g(r,s) = 1 + Sum_{k=1..r} Product_{i=0..k-1} binomial(r+s-1, s+i) / binomial(r+s-1, i).

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A005364 Hoggatt sequence with parameter d=6.

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A005365 Hoggatt sequence with parameter d=7.

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A005366 Hoggatt sequence with parameter d=8.

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A056940 Number of antichains (or order ideals) in the poset 4mn or plane partitions with at most m rows and n columns and entries <= 4.

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