A296186 Triangle read by rows, T(n,m) = (1/n)*Sum_{i=1..n} C(n,i-1)*C(n,i)*C(n,m+1-i), T(0,0)=0, m<2n.

0, 1, 1, 1, 3, 3, 1, 1, 6, 13, 13, 6, 1, 1, 10, 36, 65, 65, 36, 10, 1, 1, 15, 80, 220, 356, 356, 220, 80, 15, 1, 1, 21, 155, 595, 1380, 2072, 2072, 1380, 595, 155, 21, 1, 1, 28, 273, 1386, 4305, 8862, 12601, 12601, 8862, 4305, 1386, 273, 28, 1, 1, 36, 448, 2898, 11536, 30828, 58072, 79221, 79221, 58072, 30828, 11536

Offset

0,5

Links

Table of n, a(n) for n=0..68.

Formula

G.f.: (-sqrt((1-x*(y+1)^2)^2-4*x^2*y*(y+1)^2)-x*(y+1)^2+1)/(2*x*(y+1)*y).

Example

Triangle begins
  0;
  1,   1;
  1,   3,   3,   1;
  1,   6,  13,  13,   6,   1;
  1,  10,  36,  65,  65,  36,  10,   1;
  1,  15,  80, 220, 356, 356, 220,  80,  15,   1;

Maple

gf := (-sqrt((1-x*(y+1)^2)^2-4*x^2*y*(y+1)^2)-x*(y+1)^2+1)/(2*x*y*(y+1)):
ser := n -> series(gf, x, n+2): Y := n -> expand(simplify(coeff(ser(n), x, n))):
A296186_row := n -> `if`(n=0, [0], PolynomialTools:-CoefficientList(Y(n), y)):
ListTools:-Flatten([seq(A296186_row(n), n=0..8)]); # Peter Luschny, Jan 13 2018

Mathematica

S[n_, m_] := Binomial[n, m - 1] HypergeometricPFQ[{1 - m, 1 - n, -n }, {2, -m + n + 2}, -1]; T[n_, k_] := S[n, If[k >= n, 2 n - k + 1, k]]; Join[{{0}}, Table[T[n, k], {n, 1, 8}, {k, 1, 2 n}] ] // Flatten (* Peter Luschny, Jan 13 2018 *)
t[n_, m_] := Sum[ Binomial[n, i -1]*Binomial[n, i]*Binomial[n, m -i], {i, n}]/n;
t[0, m_] := 0; Table[t[n, m], {n, 8}, {m, 2 n}] // Flatten (* Robert G. Wilson v, Jan 22 2018 *)

Prog

(Maxima)
T(n, m):=if n=0 then 0 else 1/n*sum((binomial(n, i-1)*binomial(n, i)*binomial(n, m+1-i)), i, 1, n);

Crossrefs

Sequence in context: A086626 A244500 A300695 * A232967 A144163 A352472

Adjacent sequences: A296183 A296184 A296185 * A296187 A296188 A296189

Keyword

nonn,tabf

Author

Vladimir Kruchinin, Jan 13 2018

Status

approved