

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.


12



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
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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(n1, k) + T(n1, 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/(12x))/(12x) 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(n1, k)+T(n1, 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*Hy*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



