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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

%I

%S 1,1,1,2,5,3,5,21,28,12,14,84,180,165,55,42,330,990,1430,1001,273,132,

%T 1287,5005,10010,10920,6188,1428,429,5005,24024,61880,92820,81396,

%U 38760,7752,1430,19448,111384,352716,678300,813960,596904,245157,43263,4862

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

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

%H Jian Zhou, <a href="https://arxiv.org/abs/1810.03883">Fat and Thin Emergent Geometries of Hermitian One-Matrix Models</a>, arXiv:1810.03883 [math-ph], 2018.

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

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

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

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

%F Semidiagonal sums = A104979.

%F 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

%F 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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 5, 3;

%e 5, 21, 28, 12;

%e 14, 84, 180, 165, 55;

%e 42, 330, 990, 1430, 1001, 273;

%e 132, 1287, 5005, 10010, 10920, 6188, 1428;

%e 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752;

%e 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263; ...

%t T[n_, k_] := Binomial[2n + k, n + 2k]*Binomial[n + 2k, k]/(n + k + 1);

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-Fran├žois Alcover_, Jan 27 2019 *)

%o (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)

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%o (PARI) Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D

%o 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)

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Jun 22 2012

%o (PARI)

%o x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;

%o seq(N) = {

%o my(z0 = 1 + O((x*y)^N), z1 = 0);

%o for (k = 1, N^2,

%o z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);

%o if (z0 == z1, break()); z0 = z1);

%o vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));

%o };

%o concat(seq(9)) \\ _Gheorghe Coserea_, Nov 30 2016

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

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Mar 30 2005

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Last modified October 20 02:35 EDT 2019. Contains 328244 sequences. (Running on oeis4.)