A341867 Square array read by downward antidiagonals: T(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)*binomial(n,j)*binomial(i+j,i).

1, 2, 2, 4, 5, 4, 8, 12, 12, 8, 16, 28, 33, 28, 16, 32, 64, 86, 86, 64, 32, 64, 144, 216, 245, 216, 144, 64, 128, 320, 528, 664, 664, 528, 320, 128, 256, 704, 1264, 1736, 1921, 1736, 1264, 704, 256, 512, 1536, 2976, 4416, 5322, 5322, 4416, 2976, 1536, 512

Offset: 0

Author

Jianing Song, Nov 07 2021

Keywords

• nonn
• tabl

Comments

T(m,n) is the coefficient of x^m*y^n of 1/(1 - 2*x - 2*y + 3*x*y).

In general, define T_{s,t}(m,n) = Sum_{i=0..m, j=0..n} binomial(m,i)*binomial(n,j)*binomial(i+j,i)*s^i*t^j, then T_{s,t}(m,n) is the coefficient of x^m*y^n of 1/(1 - (1+s)*x - (1+t)*y + (1+s+t)*x*y).

T(m,n) is the coefficient of x^n of (2 - 3*x)^m/(1 - 2*x)^(m+1). In general, T_{s,t}(m,n) is the coefficient of x^n of ((1+t) - (1+s+t)*x)^m/(1 - (1+s)*x)^(m+1).

T(m,n) is odd if and only if m = n. Proof: T(m,n) == T_{-1,-1}(m,n) (mod 2). The RHS is the coefficient of x^m*y^n of 1/(1 - x*y), which is 1 if m = n and 0 otherwise.

This is the table of cardinalities of bubble posets and shuffle posets, see McConville and Mühle reference. - F. Chapoton, Sep 11 2024

Examples

			Rows 0-7:
    1,   2,    4,     8,    16,     32,     64,     128, ...
    2,   5,   12,    28,    64,    144,    320,     704, ...
    4,  12,   33,    86,   216,    528,   1264,    2976, ...
    8,  28,   86,   245,   664,   1736,   4416,   10992, ...
   16,  64,  216,   664,  1921,   5322,  14268,   37272, ...
   32, 144,  528,  1736,  5322,  15525,  43620,  118980, ...
   64, 320, 1264,  4416, 14268,  43620, 127905,  362910, ...
  128, 704, 2976, 10992, 37272, 118980, 362910, 1067925, ...
  ...
		

Links

• Jianing Song, Table of n, a(n) for n = 0..5150 (diagonals 0..100).
• T. McConville and H. Mühle, Bubble Lattices I: Structure, arXiv:2202.02874 [math.CO], 2022.
• T. McConville and H. Mühle, Bubble Lattices II: Combinatorics, arXiv:2208.13683 [math.CO], 2022.

Crossrefs

Cf. A000079 (0th row), A045623(n+1) (1st row), A343561 (2nd row), A084771 (main diagonal).

Programs

Mathematica

T[m_, n_] := Sum[Binomial[m, i] * Binomial[n, j] * Binomial[i + j, i], {i, 0, m}, {j, 0, n} ]; Table[T[m, n - m], {n, 0, 9}, {m, 0, n}] // Flatten (* Amiram Eldar, Nov 08 2021 *)
T[m_, n_] := Sum[Binomial[n, j] Hypergeometric2F1[j + 1, -m, 1, -1], {j, 0, n}];
(* Peter Luschny, Nov 08 2021 *)

PARI

T(m,n) = sum(i=0, m, sum(j=0, n, binomial(m,i)*binomial(n,j)*binomial(i+j,i)))

Formula

T(0,n) = Sum_{k=0..n} binomial(n,k) = 2^n;

T(1,n) = Sum_{k=0..n} binomial(n,k) * (k+2) = (n+4)*2^(n-1);

T(2,n) = Sum_{k=0..n} binomial(n,k) * (k^2+7*k+8)/2 = (n^2+15*n+32)*2^(n-3);

T(3,n) = Sum_{k=0..n} binomial(n,k) * (k^3+15*k^2+56*k+48)/6 = (n^3+33*n^2+254*n+384)*2^(n-4)/3.

E.g.f.: Sum_{m,n>=0} T(m,n)*x^m*y^n/(m!*n!) = exp(2*x+2*y) * BesselI(0,2*sqrt(x*y)). In general, Sum_{m,n>=0} T_{s,t}(m,n)*x^m*y^n/(m!*n!) = exp((1+s)*x+(1+t)*y) * BesselI(0,2*sqrt(s*t*x*y)). Note that BesselI(0,2*sqrt(x)) = Sum_{k>=0} x^k/(k!)^2.

E.g.f. for m-th row: Sum_{n>=0} T(m,n)*x^n/n! = exp(2*x) * Sum_{k=0..m} (binomial(m,k)*2^(m-k)/k!) * x^k. In general, Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * Sum_{k=0..m} (binomial(m,k)*(1+t)^(m-k)/k!) * (s*t*x)^k.

Define P_n(x) = exp(-x) * d^n/dx^n (x^n*exp(x)), then Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * ((1+t)^m/m!) * P_m(s*t*x/(1+t)) if t != -1 and Sum_{n>=0} T_{s,t}(m,n)*x^n/n! = exp((1+s)*x) * (s*t*x)^m/m! if t = -1.

T(m, n) = Sum_{j=0..n} binomial(n, j)*hypergeom([j + 1, -m], [1], -1). - Peter Luschny, Nov 08 2021