A104978 Triangle where g.f. satisfies: A(x,y) = 1 + x*A(x,y)^2 + x*y*A(x,y)^3, read by rows.

1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675

Offset

0,4

Comments

Row sums = A027307 (paths from (0,0) to (3n,0) in steps (2,1),(1,2),(1,-1)).

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Formula

T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).

Column 0: T(n, 0) = A000108(n) (Catalan numbers).

Main diagonal: T(n, n) = A001764(n) (ternary tree numbers).

Antidiagonal sums = A001002 (number of dissections of a polygon).

Semidiagonal sums = A104979.

G.f.: A(x,y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012

G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - Paul D. Hanna, Jun 22 2012

Example

Triangle begins:
     1;
     1,     1;
     2,     5,      3;
     5,    21,     28,     12;
    14,    84,    180,    165,     55;
    42,   330,    990,   1430,   1001,    273;
   132,  1287,   5005,  10010,  10920,   6188,   1428;
   429,  5005,  24024,  61880,  92820,  81396,  38760,   7752;
  1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263; ...

Mathematica

T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *)

Prog

(PARI) T(n, k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1, n, A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A, n, x), k, y)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D, y)); D
T(n, k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n, x), k, y)
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jun 22 2012
(PARI)
x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
seq(N) = {
  my(z0 = 1 + O((x*y)^N), z1 = 0);
  for (k = 1, N^2,
    z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
    if (z0 == z1, break()); z0 = z1);
  vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
};
concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016
(Magma) [Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 08 2021
(Sage) flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

Crossrefs

Cf. A000108, A001002, A001764, A027307, A104979.

Sequence in context: A021911 A248852 A299210 * A365723 A124568 A091807

Adjacent sequences: A104975 A104976 A104977 * A104979 A104980 A104981

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 30 2005

Status

approved