A005363 -id:A005363 - cp's OEIS frontend

A002119 Bessel polynomial y_n(-2).

1, -1, 7, -71, 1001, -18089, 398959, -10391023, 312129649, -10622799089, 403978495031, -16977719590391, 781379079653017, -39085931702241241, 2111421691000680031, -122501544009741683039, 7597207150294985028449, -501538173463478753560673

Offset: 0

N. J. A. Sloane

Absolute values give denominators of successive convergents to e using continued fraction 1+2/(1+1/(6+1/(10+1/(14+1/(18+1/(22+1/26...)))))).
Absolute values give number of different arrangements of nonnegative integers on a set of n 6-sided dice such that the dice can add to any integer from 0 to 6^n-1. For example when n=2, there are 7 arrangements that can result in any total from 0 to 35. Cf. A273013. The number of sides on the dice only needs to be the product of two distinct primes, of which 6 is the first example. - Elliott Line, Jun 10 2016
Absolute values give number of Krasner factorizations of (x^(6^n)-1)/(x-1) into n polynomials p_i(x), i=1,2,...,n, satisfying p_i(1)=6. In these expressions 6 can be replaced with any product of two distinct primes (Krasner and Ranulac, 1937). - William P. Orrick, Jan 18 2023
Absolute values give number of pairs (s, b) where s is a covering of the 1 X 2n grid with 1 X 2 dimers and equal numbers of red and blue 1 X 1 monomers and b is a bijection between the red monomers and the blue monomers that does not map adjacent monomers to each other. Ilya Gutkovskiy's formula counts such pairs by an inclusion-exclusion argument. The correspondence with Elliott Line's dice problem is that a dimer corresponds to a die containing an arithmetic progression of length 6 and a pair (r, b(r)), where r is a red monomer and b(r) its image under b, corresponds to a die containing the sum of an arithmetic progression of length 2 and an arithmetic progression of length 3. - William P. Orrick, Jan 19 2023

			Example from _William P. Orrick_, Jan 19 2023: (Start)
For n=2 the Bessel polynomial is y_2(x) = 1 + 3x + 3x^2 which satisfies y_2(-2) = -7.
The |a(2)|=7 dice pairs are
  {{0,1,2,3,4,5}, {0,6,12,18,24,30}},
  {{0,1,2,18,19,20}, {0,3,6,9,12,15}},
  {{0,1,2,9,10,11}, {0,3,6,18,21,24}},
  {{0,1,6,7,12,13}, {0,2,4,18,20,22}},
  {{0,1,12,13,24,25}, {0,2,4,6,8,10}},
  {{0,1,2,6,7,8}, {0,3,12,15,24,27}},
  {{0,1,4,5,8,9}, {0,2,12,14,24,26}}.
The corresponding Krasner factorizations of (x^36-1)/(x-1) are
  {(x^6-1)/(x-1), (x^36-1)/(x^6-1)},
  {((x^36-1)/(x^18-1))*((x^3-1)/(x-1)), (x^18-1)/(x^3-1)},
  {((x^18-1)/(x^9-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^9-1)/(x^3-1))},
  {((x^18-1)/(x^6-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^18-1))*((x^6-1)/(x^2-1))},
  {((x^36-1)/(x^12-1))*((x^2-1)/(x-1)), (x^12-1)/(x^2-1)},
  {((x^12-1)/(x^6-1))*((x^3-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^6-1)/(x^3-1))},
  {((x^12-1)/(x^4-1))*((x^2-1)/(x-1)), ((x^36-1)/(x^12-1))*((x^4-1)/(x^2-1))}.
The corresponding monomer-dimer configurations, with dimers, red monomers, and blue monomers represented by the symbols '=', 'R', and 'B', and bijections between red and blue monomers given as sets of ordered pairs, are
  (==, {}),
  (B=R, {(3,1)}),
  (BBRR, {(3,1),(4,2)}),
  (RBBR, {(1,3),(4,2)}),
  (R=B, {(1,3)}),
  (BRRB, {(2,4),(3,1)}),
  (RRBB, {(1,3),(2,4)}).
(End)

L. Euler, 1737.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270.
J. W. L. Glaisher, On Lambert's proof of the irrationality of Pi and on the irrationality of certain other quantities, Reports of British Assoc. Adv. Sci., 1871, pp. 16-18.
M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
D. H. Lehmer, Arithmetical periodicities of Bessel functions, Annals of Mathematics, 33 (1932): 143-150. The sequence but with all signs positive is on page 149.
D. H. Lehmer, Review of various tables by P. Pederson, Math. Comp., 2 (1946), 68-69.
Index entries for sequences related to Bessel functions or polynomials

Cf. A053556, A053557, A001514, A065920, A065921, A065922, A065707, A000806, A006199, A065923.
See also A033815.
Numerators of the convergents of e are A001517, which has a similar interpretation to a(n) in terms of monomer-dimer configurations, but omitting the restriction that adjacent monomers not be mapped to each other by the bijection.
Polynomial coefficients are in A001498.

f:=proc(n) option remember; if n <= 1 then 1 else f(n-2)+(4*n-2)*f(n-1); fi; end;
[seq(f(n), n=0..20)]; # This is for the unsigned version. - N. J. A. Sloane, May 09 2016
seq(simplify((-1)^n*KummerU(-n, -2*n, -1)), n = 0..17); # Peter Luschny, May 10 2022

Table[(-1)^k (2k)! Hypergeometric1F1[-k, -2k, -1]/k!, {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
nxt[{n_,a_,b_}]:={n+1,b,a-2b(2n+1)}; NestList[nxt,{1,1,-1},20][[All,2]] (* Harvey P. Dale, Aug 18 2017 *)

{a(n)= if(n<0, n=-n-1); sum(k=0, n, (2*n-k)!/ (k!*(n-k)!)* (-1)^(n-k) )} /* Michael Somos, Apr 02 2007 */

{a(n)= local(A); if(n<0, n= -n-1); A= sqrt(1 +4*x +x*O(x^n)); n!*polcoeff( exp((A-1)/2)/A, n)} /* Michael Somos, Apr 02 2007 */

{a(n)= local(A); if(n<0, n= -n-1); n+=2 ; for(k= 1, n, A+= x*O(x^k); A= truncate( (1+x)* exp(A) -1-A) ); A+= x*O(x^n); A-= A^2; -(-1)^n*n!* polcoeff( serreverse(A), n)} /* Michael Somos, Apr 02 2007 */

A002119 = lambda n: hypergeometric([-n, n+1], [], 1)
[simplify(A002119(n)) for n in (0..17)] # Peter Luschny, Oct 17 2014

D-finite with recurrence a(n) = -2(2n-1)*a(n-1) + a(n-2). - T. D. Noe, Oct 26 2006
If y = x + Sum_{k>=2} A005363(k)*x^k/k!, then y = x + Sum_{k>=2} a(k-2)(-y)^k/k!. - Michael Somos, Apr 02 2007
a(-n-1) = a(n). - Michael Somos, Apr 02 2007
a(n) = (1/n!)*Integral_{x>=-1} (-x*(1+x))^n*exp(-(1+x)). - Paul Barry, Apr 19 2010
G.f.: 1/Q(0), where Q(k) = 1 - x + 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
Expansion of exp(x) in powers of y = x*(1 + x): exp(x) = 1 + y - y^2/2! + 7*y^3/3! - 71*y^4/4! + 1001*y^5/5! - .... E.g.f.: (1/sqrt(4*x + 1))*exp(sqrt(4*x + 1)/2 - 1/2) = 1 - x + 7*x^2/2! - 71*x^3/3! + .... - Peter Bala, Dec 15 2013
a(n) = hypergeom([-n, n+1], [], 1). - Peter Luschny, Oct 17 2014
a(n) = sqrt(Pi/exp(1)) * BesselI(1/2+n, 1/2) + (-1)^n * BesselK(1/2+n, 1/2) / sqrt(exp(1)*Pi). - Vaclav Kotesovec, Jul 22 2015
a(n) ~ (-1)^n * 2^(2*n+1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Jul 22 2015
From G. C. Greubel, Aug 16 2017: (Start)
G.f.: (1/(1-t))*hypergeometric2f0(1, 1/2; -; -4*t/(1-t)^2).
E.g.f.: (1+4*t)^(-1/2) * exp((sqrt(1+4*t) - 1)/2). (End)
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*binomial(n+k,k)*k!. - Ilya Gutkovskiy, Nov 24 2017
a(n) = (-1)^n*KummerU(-n, -2*n, -1). - Peter Luschny, May 10 2022

More terms from Vladeta Jovovic, Apr 03 2000

A056941 Number of antichains (or order ideals) in the poset 5mn or plane partitions with not more than m rows, n columns and entries <= 5.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 21, 21, 1, 1, 56, 196, 56, 1, 1, 126, 1176, 1176, 126, 1, 1, 252, 5292, 14112, 5292, 252, 1, 1, 462, 19404, 116424, 116424, 19404, 462, 1, 1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1, 1, 1287, 169884, 3737448, 16818516, 16818516, 3737448, 169884, 1287, 1

Offset: 0

Mitch Harris

Triangle of generalized binomial coefficients (n,k)A342889.%20-%20_N.%20J.%20A.%20Sloane">5; cf. A342889. - _N. J. A. Sloane, Apr 03 2021

			The array starts:
  [1    1      1        1          1           1            1 ...]
  [1    6     21       56        126         252          462 ...]
  [1   21    196     1176       5292       19404        60984 ...]
  [1   56   1176    14112     116424      731808      3737448 ...]
  [1  126   5292   116424    1646568    16818516    133613766 ...]
  [1  252  19404   731808   16818516   267227532   3184461423 ...]
  [1  462  60984  3737448  133613766  3184461423  55197331332 ...]
  [...]
Considered as a triangle, the initial rows are:
   1;
   1,   1;
   1,   6,     1;
   1,  21,    21,      1;
   1,  56,   196,     56,       1;
   1, 126,  1176,   1176,     126,      1;
   1, 252,  5292,  14112,    5292,    252,     1;
   1, 462, 19404, 116424,  116424,  19404,   462,   1;
   1, 792, 60984, 731808, 1646568, 731808, 60984, 792, 1; ...

Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
P. A. MacMahon, Combinatory Analysis, Section 495, 1916.
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

Seiichi Manyama, Rows n = 0..139 of triangle, flattened
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.
Index entries for sequences related to posets

Cf. A000372, A001263, A005363 (row sums), A056932, A056939, A056940, A142465, A142467, A142468.
Antidiagonals sum to A005363 (Hoggatt sequence).
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.

A056941:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..4]]) >;
[A056941(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 14 2022

T[n_, k_] := Product[Binomial[n+j, k]/Binomial[k+j, k], {j,0,4}];
Table[T[n, k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 14 2022 *)

A056941(n,m)=prod(k=0,4,binomial(n+m+k,m+k)/binomial(n+k,k)) \\ as an array \\ M. F. Hasler, Sep 26 2018

def A056941(n,k): return product(binomial(n+j,k)/binomial(k+j,k) for j in (0..4))
flatten([[A056941(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Nov 14 2022

From Peter Bala, Oct 13 2011: (Start)
A(n, k) = Product_{j=0..4} C(n+k+j, k+j)/C(n+j, j) gives the array as a square.
g(n-1, k-1)*g(n, k+1)*g(n+1, k) = g(n-1, k)*g(n, k-1)*g(n+1, k+1) where g(n, k) is the array A(n, k) and triangle T(n, k).
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)
From Peter Bala, May 10 2012: (Start)
Determinants of 5 X 5 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present).
Also determinants of 5 X 5 arrays whose entries come from a single row:
  det [C(n,k), C(n,k-1), C(n,k-2), C(n,k-3), C(n,k-4); C(n,k+1), C(n,k), C(n,k-1), C(n,k-2), C(n,k-3); C(n,k+2), C(n,k+1), C(n,k), C(n,k-1), C(n,k-2); C(n,k+3), C(n,k+2), C(n,k+1), C(n,k), C(n,k-1); C(n,k+4), C(n,k+3), C(n,k+2), C(n,k+1), C(n,k)]. (End)
From G. C. Greubel, Nov 14 2022: (Start)
T(n, k) = Product_{j=0..4} binomial(n+j, k)/binomial(k+j, k) (gives the triangle).
Sum_{k=0..n} T(n, k) = A005363(n). (End)

Edited by M. F. Hasler, Sep 26 2018

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

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A002119 Bessel polynomial y_n(-2).

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A056941 Number of antichains (or order ideals) in the poset 5*m*n or plane partitions with not more than m rows, n columns and entries <= 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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A056941 Number of antichains (or order ideals) in the poset 5mn or plane partitions with not more than m rows, n columns and entries <= 5.

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