A092686 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, when flattened, equals this flattened form of the original 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, 1120, 3278, 3056, 2900, 2776, 2704, 2592, 2656, 2240, 3278, 9612, 9012, 8576, 8256, 8000, 7840, 7552, 7758

Offset

0,2

Comments

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 polynomial 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, Jul 17 2006

Links

Paul D. Hanna, Table of n, a(n) for n = 0..495

Formula

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, Jul 17 2006

Example

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;
28236, 26600, 25408, 24512, 23840, 23232, 22862, 22072, 22724, 19224, 28236; ...
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.

Prog

(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)))))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(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)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Jul 17 2006

Crossrefs

Cf. A092683, A092687, A092688, A092689, A120894, A120898.

Sequence in context: A273012 A222404 A081111 * A249796 A182411 A067804

Adjacent sequences: A092683 A092684 A092685 * A092687 A092688 A092689

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 04 2004

Status

approved