A001181 -id:A001181 - cp's OEIS frontend

A046996 Number of Baxter permutations: A001181/2.

1, 3, 11, 46, 211, 1037, 5377, 29101, 163120, 941480, 5570280, 33664996, 207249719, 1296670793, 8229378293, 52895993341, 343891293422, 2258771535962, 14974619271658, 100117092310368, 674548712552456, 4577138309318008, 31261253291922136, 214800030086785976

Offset: 2

N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 2..1120

Cf. A359363.

a:= proc(n) option remember; `if`(n<4, 2*n-3,
      ((7*n^2+7*n-2)*a(n-1)+8*(n-1)*(n-2)*a(n-2))/((n+2)*(n+3)))
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Jul 29 2022

Sum[Binomial[n+1, k-1] Binomial[n+1, k] Binomial[n+1, k+1], {k, 1, n}]/(2 Binomial[n+1, 1] Binomial[n+1, 2])

def A046996(n): return PolyA359363(n, 1) // 2
print([A046996(n) for n in range(2, 26)])  # Peter Luschny, Jan 04 2024

a(n) = (1/2)*B(n, 1) where B(n, x) are the Baxter polynomials with coefficients A359363. - Peter Luschny, Jan 04 2024

More terms from Olivier Gérard, Jun 26 2001

A342282 G.f. = 1-1/H(x) where H(x) = 1 + 2x + 6x^2 + 22x^3 + 92x^4 + 422x^5 + 2074x^6 + ... is a g.f. for the Baxter sequence A001181 with a different offset.

Original entry on oeis.org

0, 2, 2, 6, 24, 110, 550, 2922, 16242, 93520, 553980, 3359384, 20777588, 130696662, 834244830, 5393850898, 35272830054, 233016356788, 1553427829684, 10441803227652, 70715551631992, 482201548554776, 3308810614160224, 22836540660981088, 158458108961055864, 1104984826809182592

Offset: 0

N. J. A. Sloane, Mar 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Essam, J.W. and Guttmann, A.J., 1995. Vicious walkers and directed polymer networks in general dimensions. Physical Review E, 52(6), pp. 5849ff.
Iwan Jensen, Three friendly walkers, Journal of Physics A: Mathematical and Theoretical, Volume 50:2 (2017), #24003, 14 pages. See G(x).

Cf. A001181, A342283, A342284.

A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.

Original entry on oeis.org

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

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20This%20array%20is%20the%20main%20subject%20of%20the%20long%20article%20by%20Felsner%20et%20al.%20(2011).%20-%20_N.%20J.%20A.%20Sloane">3; cf. A342889. This array is the main subject of the long article by Felsner et al. (2011). - _N. J. A. Sloane, Apr 03 2021
This triangle is mentioned by Hoggatt (1977). - N. J. A. Sloane, Mar 27 2021
Determinants of 3 X 3 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005
Also determinants of 3 X 3 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+1),C(n,k),C(n,k-1); C(n,k+2),C(n,k+1),C(n,k)]. - Peter Bala, May 10 2012
From Gary W. Adamson, Jul 10 2012: (Start)
The triangular view of this triangle is
  1;
  1,  1;
  1,  4,  1;
  1, 10, 10,  1;
  1, 20, 50, 20,  1;
The n-th row of this triangle is generated by applying the ConvOffs transform to the first n terms of 1, 4, 10, 20, ... (A000292 without leading zero). See A214281 for a procedural definition of the transformation and search "ConvOffs" for more examples. (End)
Define polynomials p(n, x) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -x). If the triangle is extended by the diagonal 1, 0, 0,... on the right side the resulting (0, 0)-based triangle is T*(n, k) = [x^k] p(n, x). The polynomials evaluated at x = 1 gives the number of Baxter permutations of length n (see the formula given by Richard L. Ollerton in A001181). - Peter Luschny, Dec 28 2022

			The initial rows of the array are:
     1      1      1      1      1      1 ...
     1      4     10     20     35     56 ...
     1     10     50    175    490   1176 ...
     1     20    175    980   4116  14112 ...
     1     35    490   4116  24696 116424 ...
     1     56   1176  14112 116424 731808 ...
     ...
Considered as a triangle, the initial rows are:
  [1],
  [1, 1],
  [1, 4, 1],
  [1, 10, 10, 1],
  [1, 20, 50, 20, 1],
  [1, 35, 175, 175, 35, 1],
  [1, 56, 490, 980, 490, 56, 1],
  [1, 84, 1176, 4116, 4116, 1176, 84, 1],
  [1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1],
  [1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1],
  [1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1]
  ...

Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

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).
Johann Cigler, Some observations about Hankel determinants of the columns of Pascal triangle and related topics, arXiv:2202.07298 [math.CO], 2022.
Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
P. A. MacMahon, Combinatory analysis, section 495, 1916.

Cf. A000372, A056932, A001263, A056940, A056941.
Antidiagonals sum to A001181 (Baxter permutations). Cf. A197208.
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.

# To get initial terms of the array - N. J. A. Sloane, Apr 20 2021
bb := (k,l) -> binomial(k+l,k)*binomial(k+l+1,k)*binomial(k+l+2,k)*2/((k+1)^2*(k+2));
for k from 0 to 8 do
lprint([seq(bb(k,l),l=0..8)]);
od:

t[n_, m_] = 2*Binomial[n, m]*Binomial[n + 1, m + 1]* Binomial[n + 2, m + 2]/((n - m + 1)^2*(n - m + 2)); Flatten[Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]] (* Roger L. Bagula, Jan 28 2009 *)

\\ cf. A359363
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=1,10,for(k=1,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 04 2024

Product_{k=0..2} binomial(n+m+k, m+k)/binomial(n+k, k) gives the array as a square.
T(n,m) = 2*binomial(n, m)*binomial(n+1, m+1)*binomial(n+2, m+2)/((n-m+1)^2*(n-m+2)). - Roger L. Bagula, Jan 28 2009
From Peter Bala, Oct 13 2011: (Start)
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1);
T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k+1)*C(n+2,k)*C(n+3,k+2). Cf. A197208.
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 a(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is a(r,0)*a(r,n)/(a(r,k)*a(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)
The column generating functions of the square array (starting at column 1) are 1/(1 - x)^4, (1 + 3*x + x^2)/(1 - x)^7, (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^10, ..., where the numerator polynomials are the row polynomials of A087647. See Barry p. 31. - Peter Bala, Oct 18 2023

A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, 5583, 19683, 72162, 274796, 1082439, 4406706, 18484332, 79818616, 353995743, 1611041726, 7510754022, 35842380314, 174850257639, 871343536591, 4430997592209, 22978251206350, 121410382810005, 653225968918521

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 13 2014

nonn

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
Also the number of ballot sequences without two consecutive elements equal. A ballot sequence B is a string such that, for all prefixes P of B, h(i)>=h(j) for iA000085).
First column (k=0) of A238125.

			The a(5) = 9 such tableaux of 5 are:
[1]   [2]  [3]   [4]  [5]  [6]  [7]  [8]  [9]
135   13   135   13   13   14   14   15   1
24    24   2     25   2    25   2    2    2
      5    4     4    4    3    3    3    3
                      5         5    4    4
                                          5
The corresponding ballot sequences are:
1:  [ 0 1 0 1 0 ]
2:  [ 0 1 0 1 2 ]
3:  [ 0 1 0 2 0 ]
4:  [ 0 1 0 2 1 ]
5:  [ 0 1 0 2 3 ]
6:  [ 0 1 2 0 1 ]
7:  [ 0 1 2 0 3 ]
8:  [ 0 1 2 3 0 ]
9:  [ 0 1 2 3 4 ]

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..68 (terms 0..48 from Alois P. Heinz)
Timothy Y. Chow, Henrik Eriksson and C. Kenneth Fan, Chess Tableaux, The Electronic Journal of Combinatorics, vol.11, no.2, (2005).
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Cf. A000085 (all Young tableaux), A000957, A001181, A214021, A214087, A214159, A214875.
Cf. A238126 (tableaux with one succession), A238127 (two successions).
Cf. A264051, A264078.

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [i, l[]]))))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=0..30);
# second Maple program (counting ballot sequences):
b:= proc(n, v, l) option remember;
      `if`(n<1, 1, add(`if`(i<>v and (i=1 or l[i-1]>l[i]),
       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
       b(n-1, nops(l)+1, [l[], 1]))
    end:
a:= proc(n) option remember; forget(b); b(n-1, 1, [1]) end:
seq(a(n), n=0..30);

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i != v && (i == 1 || l[[i-1]] > l[[i]]), b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; a[n_] := a[n] = b[n-1, 1, {1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, translated from 2nd Maple program *)

a(n) = Sum_{k=1..A264078(n)} k * A264051(n,k). - Alois P. Heinz, Nov 02 2015

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

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

A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 6, 6, 1, 1, 1, 0, 1, 22, 72, 18, 1, 1, 1, 0, 1, 92, 1289, 960, 57, 1, 1, 1, 0, 1, 422, 29889, 93964, 14257, 186, 1, 1, 1, 0, 1, 2074, 831174, 13652068, 8203915, 228738, 622, 1, 1

Offset: 0

Alois P. Heinz, Jul 01 2012

A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.

			A(2,4) = 1:
  [1 3 5 7]
  [2 4 6 8].
A(4,2) = 6:
  [1, 5]   [1, 4]   [1, 3]   [1, 4]   [1, 3]   [1, 3]
  [2, 6]   [2, 6]   [2, 6]   [2, 5]   [2, 5]   [2, 4]
  [3, 7]   [3, 7]   [4, 7]   [3, 7]   [4, 7]   [5, 7]
  [4, 8]   [5, 8]   [5, 8]   [6, 8]   [6, 8]   [6, 8].
Square array A(n,k) begins:
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  0,     0,       0,          0,              0, ...
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  2,     6,      22,         92,            422, ...
  1, 1,  6,    72,    1289,      29889,         831174, ...
  1, 1, 18,   960,   93964,   13652068,     2621897048, ...
  1, 1, 57, 14257, 8203915, 8134044455, 11865331748843, ...

Alois P. Heinz, Antidiagonals n = 0..23, flattened
T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
Wikipedia, Young tableau

Rows n=0+2, 3-4 give: A000012, A001181(k) for k>0, A214875.
Columns k=0+1, 2, 3 give: A000012, A000957(n+1), A214159.
Main diagonal gives A264103.
Cf. A214020, A214088.

b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
       add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
    end:
A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, l}]}; If[s == 0, 1, Sum[If[t != i && l[[i]] > If[i == n, 0, l[[i+1]]], b[ReplacePart[l, i -> l[[i]]-1], i], 0], {i, 1, n}]] ] ; a[n_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

cp's OEIS Frontend

A046996 Number of Baxter permutations: A001181/2.

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A342282 G.f. = 1-1/H(x) where H(x) = 1 + 2*x + 6*x^2 + 22*x^3 + 92*x^4 + 422*x^5 + 2074*x^6 + ... is a g.f. for the Baxter sequence A001181 with a different offset.

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A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.

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A237770 Number of standard Young tableaux with n cells without a succession v, v+1 in a row.

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A005362 Hoggatt sequence with parameter d=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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A342282 G.f. = 1-1/H(x) where H(x) = 1 + 2x + 6x^2 + 22x^3 + 92x^4 + 422x^5 + 2074x^6 + ... is a g.f. for the Baxter sequence A001181 with a different offset.

A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3mn or plane partitions with rows <= m, columns <= n and entries <= 3.