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 A104978 Triangle where g.f. satisfies: A(x,y) = 1 + x*A(x,y)^2 + x*y*A(x,y)^3, read by rows. 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Row sums = A027307 (paths from (0,0) to (3n,0) in steps (2,1),(1,2),(1,-1)). LINKS 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, 8}, {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 CROSSREFS Cf. A000108, A001764, A027307, A001002, A104979. Sequence in context: A021911 A248852 A299210 * A124568 A091807 A085825 Adjacent sequences:  A104975 A104976 A104977 * A104979 A104980 A104981 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Mar 30 2005 STATUS approved

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Last modified May 12 03:08 EDT 2021. Contains 343810 sequences. (Running on oeis4.)