0,2

First column and main diagonal forms A092687. Row sums form A092688.

This triangle is the cascadence of binomial (1+2x). More generally, the cascadence of polyomial F(x) of degree d, F(0)=1, is a triangle with d*n+1 terms in row n where the g.f. of the triangle, A(x,y), is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) satisfies: H(x) = G*H(x*G^d)/x, and G=G(x) satisfies: G(x) = x*F(G(x)) so that G = series_reversion(x/F(x)); also, H(x) is the g.f. of column 0. - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006

T(n, k) = 2*T(n-1, k) + T(n-1, k+1) for 0=<k<n, with T(n, n)=T(n, 0), T(0, 0)=1, T(0, 1)=T(1, 0)=2.

G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+2y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-2x))/(1-2x), and H(x) is the g.f. of column 0 (A092687). - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006

Rows begin:

{1},

{2,2},

{6,4,6},

{16,14,12,16},

{46,40,40,32,46},

{132,120,112,110,92,132},

{384,352,334,312,316,264,384},

{1120,1038,980,940,896,912,768,1120},

{3278,3056,2900,2776,2704,2592,2656,2240,3278},

{9612,9012,8576,8256,8000,7840,7552,7758,6556,9612},...

Convolution of each row with {1,2} results in the triangle:

{1,2},

{2,6,4},

{6,16,14,12},

{16,46,40,40,32},

{46,132,120,112,110,92},

{132,384,352,334,312,316,264},

{384,1120,1038,980,940,896,912,768},...

which, when flattened, equals the original triangle in flattened form.

(PARI) {T(n, k)=if(n<0|k>n, 0, if(n==0&k==0, 1, if(n==1&k<=1, 2, if(k==n, T(n, 0), 2*T(n-1, k)+T(n-1, k+1)))))}

(PARI) /* Generate Triangle by the G.F.: */ {T(n, k)=local(A, F=1+2*x, d=1, G=x, H=1+2*x, S=ceil(log(n+1)/log(d+1))); for(i=0, n, G=x*subst(F, x, G+x*O(x^n))); for(i=0, S, H=subst(H, x, x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H, x, x*y^d +x*O(x^n)))/(x*subst(F, x, y)-y); polcoeff(polcoeff(A, n, x), k, y)} - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006

Cf. A092683, A092687, A092688, A092689.

Cf. A120894, A120898.

Sequence in context: A036500 A077080 A081111 this_sequence A067804 A074911 A071059

Adjacent sequences: A092683 A092684 A092685 this_sequence A092687 A092688 A092689

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 04 2004