A047080 -id:A047080 - cp's OEIS frontend

A047081 a(n) = Sum_{k=0..n} T(n, k), array T as in A047080.

1, 2, 3, 6, 11, 20, 37, 68, 125, 230, 423, 778, 1431, 2632, 4841, 8904, 16377, 30122, 55403, 101902, 187427, 344732, 634061, 1166220, 2145013, 3945294, 7256527, 13346834, 24548655, 45152016, 83047505, 152748176, 280947697, 516743378, 950439251, 1748130326, 3215312955, 5913882532, 10877325813

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A047080, A047082, A047083, A047084, A047085, A047086, A047087, A047088.

Cf. A000073, A254006.

[n le 3 select n else Self(n-1) +Self(n-2) +Self(n-3): n in [1..60]]; // G. C. Greubel, Oct 31 2022

LinearRecurrence[{1,1,1}, {1,2,3}, 61] (* G. C. Greubel, Oct 31 2022 *)

@CachedFunction
def a(n): return (n+1) if (n<3) else a(n-1) +a(n-2) +a(n-3) # a = A047081
[a(n) for n in (0..60)] # G. C. Greubel, Oct 31 2022

From G. C. Greubel, Oct 31 2022: (Start)
G.f.: (1 + x)/(1 - x - x^2 - x^3).
a(n) = A000073(n+1) + A000073(n+2). (End)
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-3*r^2+5*r+2). - Fabian Pereyra, Nov 23 2024
a(n) = A001590(n+3). - R. J. Mathar, Mar 28 2025

Data corrected by G. C. Greubel, Oct 31 2022

A047085 a(n) = T(2*n, n), array T as in A047080.

Original entry on oeis.org

1, 1, 3, 9, 27, 83, 259, 817, 2599, 8323, 26797, 86659, 281287, 915907, 2990383, 9786369, 32092959, 105435607, 346950321, 1143342603, 3772698725, 12463525229, 41218894577, 136451431723, 452116980643, 1499282161375, 4975631425581, 16524213199923, 54913514061867

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A047080, A047081, A047082, A047083, A047084, A047086, A047087, A047088.

Cf. A171155.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( Sqrt((1-x)/(1 -3*x-x^2-x^3)) )); // G. C. Greubel, Oct 30 2022

CoefficientList[Series[Sqrt[(1-x)/(1-3*x-x^2-x^3)], {x, 0, 50}], x] (* G. C. Greubel, Oct 30 2022 *)

def A047085_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-3*x-x^2-x^3)) ).list()
A047085_list(50) # G. C. Greubel, Oct 30 2022

From G. C. Greubel, Oct 30 2022: (Start)
a(n) = A171155(n).
G.f.: sqrt((1 - x)/(1 - 3*x - x^2 - x^3)). (End)

Data corrected by Sean A. Irvine, May 11 2021

A047082 a(n) = Sum_{i=0..floor(n/2)} A047080(n,i).

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 23, 34, 76, 115, 253, 389, 845, 1316, 2829, 4452, 9488, 15061, 31863, 50951, 107112, 172366, 360360, 583110, 1213150, 1972647, 4086217, 6673417, 13769519, 22576008, 46416937, 76374088, 156520328, 258371689, 527937429, 874065163, 1781131638

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A047080, A047081, A047083, A047084, A047085, A047086, A047087, A047088.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[(&+[A(n-j,j): j in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, Oct 31 2022

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] - Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j,0,Floor[(n+k-2)/3]}];
A047082[n_]:= A047082[n]= Sum[A[n-k,k], {k,0,Floor[n/2]}];
Table[A047082[n], {n, 0, 50}] (* G. C. Greubel, Oct 31 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[sum(A(n-j,j) for j in range(1+(n//2))) for n in range(51)] # G. C. Greubel, Oct 31 2022

Data corrected by Sean A. Irvine, May 11 2021

A047083 a(n) = Sum_{i=0..floor((n+1)/2)} A047080(n,i).

Original entry on oeis.org

1, 2, 2, 5, 7, 15, 23, 49, 76, 161, 253, 532, 845, 1766, 2829, 5881, 9488, 19631, 31863, 65649, 107112, 219857, 360360, 737152, 1213150, 2473930, 4086217, 8309252, 13769519, 27927146, 46416937, 93915759, 156520328, 315982677, 527937429, 1063586803, 1781131638

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A047080, A047081, A047082, A047084, A047085, A047086, A047087, A047088.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[(&+[A(n-j,j): j in [0..Floor((n+1)/2)]]): n in [0..50]]; // G. C. Greubel, Oct 31 2022

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -
 Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j,0,Floor[(n+k-2)/3]}];
A047083[n_]:= A047083[n]= Sum[A[n-k,k], {k,0,Floor[(n+1)/2]}];
Table[A047083[n], {n,0,50}] (* G. C. Greubel, Oct 31 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[sum(A(n-j,j) for j in range(1+((n+1)//2))) for n in range(51)] # G. C. Greubel, Oct 31 2022

Data corrected by Sean A. Irvine, May 11 2021

A047084 a(n) = Sum_{i=0..n} A047080(i,n-i).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 14, 21, 33, 50, 77, 118, 181, 278, 426, 654, 1003, 1539, 2361, 3622, 5557, 8525, 13079, 20065, 30783, 47226, 72452, 111153, 170526, 261614, 401357, 615745, 944650, 1449242, 2223366, 3410994, 5233003, 8028252, 12316605, 18895615, 28988854

Offset: 0

Clark Kimberling

Sean A. Irvine, Table of n, a(n) for n = 0..1000

Cf. A047080, A047081, A047082, A047083, A047085, A047086, A047087, A047088.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[(&+[A(n-2*j, j): j in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, Oct 31 2022

A[n_, k_]:=Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -
 Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j,0,Floor[(n+k-2)/3]}];
A047084[n_]:= A047084[n]= Sum[A[2*k-n, n-k], {k,0,n}];
Table[A047084[n], {n, 0, 50}] (* G. C. Greubel, Oct 31 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[sum(A(n-2*j,j) for j in range(1+(n//2))) for n in range(51)] # G. C. Greubel, Oct 31 2022

a(n) = Sum_{j=0..floor(n/2)} A(n-2*j, j), where A(n,k) = array of A048080(n,k). - G. C. Greubel, Oct 31 2022

Entry revised by Sean A. Irvine, May 11 2021

A047086 a(n) = T(2*n+1, n), array T as in A047080.

Original entry on oeis.org

1, 2, 5, 15, 46, 143, 450, 1429, 4570, 14698, 47491, 154042, 501283, 1635835, 5351138, 17541671, 57610988, 189521640, 624389105, 2059824523, 6803433916, 22495796651, 74457478476, 246667937610, 817866796549, 2713874203112, 9011747680649, 29944572743724

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047087, A047088.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[A(n+1,n): n in [0..50]]; // G. C. Greubel, Oct 30 2022

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j, 0, Floor[(n+k)/3]}] - Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j, 0, Floor[(n+k-2)/3]}];
T[n_, k_]:= A[n-k,k];
Table[T[2*n+1,n], {n,0,50}] (* G. C. Greubel, Oct 30 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[A(n+1,n) for n in range(51)] # G. C. Greubel, Oct 30 2022

a(n+4) = ((16*n^3 + 100*n^2 + 188*n + 105)*a(n+3) - (8*n^3 + 36*n^2 + 46*n + 5)*a(n+2) + (4*n^2 + 16*n + 25)*a(n+1) - (n-1)*(2*n+5)^2*a(n))/((n+4)*(2*n+3)^2). - G. C. Greubel, Oct 30 2022

Corrected and extended by Sean A. Irvine, May 11 2021

A047087 a(n) = A047080(2*n, n+1).

Original entry on oeis.org

1, 3, 8, 24, 75, 237, 755, 2421, 7804, 25264, 82081, 267487, 873970, 2862038, 9391137, 30869167, 101627704, 335049772, 1106003560, 3655124296, 12092095945, 40042017815, 132712302538, 440207294382, 1461259979347, 4853983051617, 16134233746913, 53660996850207

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, A047088.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[A(n-1,n+1): n in [1..50]]; // G. C. Greubel, Oct 30 2022

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] - Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j, 0, Floor[(n+k- 2)/3]}];
Table[A[n-1, n+1], {n, 50}] (* G. C. Greubel, Oct 30 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[A(n-1,n+1) for n in range(1,50)] # G. C. Greubel, Oct 30 2022

a(n+4) = ((4*n^5 + 61*n^4 + 374*n^3 + 1146*n^2 + 1743*n + 1046)*a(n+3) - (2*n^5 + 27*n^4 + 146*n^3 + 380*n^2 + 467*n + 220)*a(n+2) + (n+4)*(n^3 + 10*n^2 + 44*n + 53)*a(n+1) - (n-2)*(n+3)*(n+4)*(n^2 + 8*n + 18)*a(n))/((n+2)*(n+3)*(n+5)*(n^2 + 6*n + 11)). - G. C. Greubel, Oct 30 2022

Corrected and extended by Sean A. Irvine, May 11 2021

A047088 a(n) = A047080(2*n+1, n+2).

Original entry on oeis.org

1, 4, 12, 37, 118, 380, 1229, 3989, 12987, 42394, 138709, 454768, 1493690, 4913969, 16189534, 53407853, 176397299, 583242159, 1930349545, 6394665589, 21201345460, 70346920007, 233581374587, 776105485336, 2580316142887, 8583746045611, 28570407158100

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, A047087.

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[A(n-1,n+2): n in [1..50]]; // G. C. Greubel, Oct 31 2022

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] - Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j,0,Floor[(n+k-2)/3]}];
Table[A[n-1, n+2], {n, 50}] (* G. C. Greubel, Oct 31 2022 *)

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[A(n-1,n+2) for n in range(1,50)] # G. C. Greubel, Oct 31 2022

a(n+4) = ((16*n^5 + 324*n^4 + 2624*n^3 + 10509*n^2 + 20655*n + 15930)*a(n+3) - (8*n^5 + 148*n^4 + 1090*n^3 + 3953*n^2 + 7365*n + 5994)*a(n+2) + (4*n^4 + 84*n^3 + 701*n^2 + 2451*n + 2646)*a(n+1) - (n-3)*(n+6)*(2*n+7)*(2*n^2 + 23*n + 72)*a(n) )/((n+3)*(n+6)*(2*n+5)*(2*n^2 + 19*n + 51)). - G. C. Greubel, Oct 31 2022

Corrected and extended by Sean A. Irvine, May 11 2021

A180091 a(m,n) is the number of ways to split two strings of length m and n, respectively, into the same number of nonempty parts such that at least one of the corresponding parts has length 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 9, 1, 4, 8, 15, 27, 1, 5, 12, 24, 46, 83, 1, 6, 17, 37, 75, 143, 259, 1, 7, 23, 55, 118, 237, 450, 817, 1, 8, 30, 79, 180, 380, 755, 1429, 2599, 1, 9, 38, 110, 267, 592, 1229, 2421, 4570, 8323

Offset: 1

Steffen Eger, Jan 14 2011

a(m,n) is also the number alignments (between two strings) that satisfy weak monotonicity, completeness, and disjointness.
a(m,n) is also the number of monotone lattice paths from (0,0) to (m,n) with steps in {(1,1),(1,2),(1,3),(1,4),...,(2,1),(3,1),(4,1),...}. - Steffen Eger, Sep 25 2012
From Julien Rouyer, Jun 02 2023: (Start)
a(m-1,n-1) is also the number of strictly increasing functions defined on a part of the m-set {1,...,m} that take values in the n-set {1,...,n} and that can't be extended to a greater part of the m-set to give another strictly increasing function (values for m < n can be obtained by symmetry).
a(m-1,n-1) is also the number of solutions to the problem consisting of connecting, with noncrossing edges, some of m points aligned on a straight line to some of n other points aligned on a parallel straight line (each point is connected at most with one other point), in such a way that no additional noncrossing connection can be added.
A direct combinatorial calculation is possible, but time-consuming if m and n are large. (End)
From Thierry Marchant, Oct 30 2024: (Start)
a(m,n) is also the number of maximal antichains in the product of two chains of length m and n.
a(m,n) is also the number of strict chains in the product of two chains of length m and n (a strict chain P in a product of two chains is a chain such that x,y in P implies x_1 different from y_1 and x_2 different from y_2).
a(m,n) is also the number of walks from (0,0) to (m,n) where unit horizontal (+1,0), vertical (0,+1), and diagonal (+1,+1) steps are permitted but a horizontal step cannot be followed by a vertical step, nor vice versa. (End)

			For m=4, n=3, the 5 possibilities are:
a) X XXX   b) XXX X  c) X XX X  d) XX X X   e) X X XX
   YY Y        Y YY     Y  Y Y     Y  Y Y      Y Y Y
The triangle a(m,n) starts in row m=1 with columns 1 <= n <= m as:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  5,   9;
  1,  4,  8,  15,  27;
  1,  5, 12,  24,  46,   83;
  1,  6, 17,  37,  75,  143,  259;
  1,  7, 23,  55, 118,  237,  450,  817;
  1,  8, 30,  79, 180,  380,  755, 1429,  2599;
  1,  9, 38, 110, 267,  592, 1229, 2421,  4570,  8323;
  1, 10, 47, 149, 386,  899, 1948, 3989,  7804, 14698, 26797;
  1, 11, 57, 197, 545, 1334, 3015, 6412, 12987, 25264, 47491, 86659;
From _Julien Rouyer_, Jun 02 2023: (Start)
There are a(5)=T(3,2)=5 strictly increasing functions defined on a part of {1,2,3} that take values in {1,2} and can't be extended keeping the same properties. The 5 functions are defined by
    f(1)=1, f(2)=2;
    g(1)=1, g(2)=3;
    h(1)=2, h(2)=3;
    i(1)=3;
    j(2)=1. (End)

D. Bouyssou, T. Marchant, and M. Pirlot, About maximal antichains in a product of two chains:A catch-all note, arXiv:2410.16243 [math.CO], 2024. See pp. 1, 3, 16-18.
M. A. Covington, The Number of Distinct Alignments of Two Strings, Journal of Quantitative Linguistics, Vol. 11 (2004), Issue 3, pp. 173-182.
S. Eger, Derivation of sequence [broken link]

Cf. A089071 (third column), A108626 (sums of diagonals).
Main diagonal gives A171155.
Cf. A047080.

A180091 := proc(m,n) a := binomial(m-1,n-1); for k from 2 to n-1 do for l from 1 to k-1 do if k-l-1 >= 0 and k-l-1 <= n-k-1 and l-1 >=0 and l-1 <= m+l-k-1 then a := a+ binomial(k,l)*binomial(n-k-1,k-l-1)*binomial(m+l-k-1,l-1); end if; end do: end do: a; end proc: # R. J. Mathar, Feb 01 2011

# The following program gives T(m,n)=a(m+1,n+1) for any m >= 0 and n >= 0:
def T(m,n):
    if n == 0:
        res = 1
    elif n == 1:
        res = max(m,n)
    elif m < n:
        res = T(n,m)
    else:
        res = 0
        for i in range(1,m+1):
            res += T(m-i,n-1)
        for j in range(2,n+1):
            res += T(m-1,n-j)
    return res # Julien Rouyer, Jun 02 2023

For m >= n: a(m,n) = C(m-1,n-1) + Sum_{k=2..n-1} Sum_{i=1..k-1} C(k,i)*C(n-k-1,k-i-1)*C(m+i-k-1,i-1).
Symmetrically extended to m <= n by a(m,n) = a(n,m).
a(n,n) = A171155(n-1).
a(m,n) = Sum_{i=1..m} a(m-i,n-1) + Sum_{j=2..n} a(m-1,n-j). - Julien Rouyer, Jun 02 2023

cp's OEIS Frontend

A047081 a(n) = Sum_{k=0..n} T(n, k), array T as in A047080.

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A047085 a(n) = T(2*n, n), array T as in A047080.

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A047082 a(n) = Sum_{i=0..floor(n/2)} A047080(n,i).

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A047083 a(n) = Sum_{i=0..floor((n+1)/2)} A047080(n,i).

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A047084 a(n) = Sum_{i=0..n} A047080(i,n-i).

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A047086 a(n) = T(2*n+1, n), array T as in A047080.

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A047087 a(n) = A047080(2*n, n+1).

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