A092683 Triangle, read by rows, such that the convolution of each row with {1,1} produces a triangle which, when flattened, equals this flattened form of the original triangle.

1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 6, 5, 5, 3, 6, 11, 10, 8, 9, 6, 11, 21, 18, 17, 15, 17, 11, 21, 39, 35, 32, 32, 28, 32, 21, 39, 74, 67, 64, 60, 60, 53, 60, 39, 74, 141, 131, 124, 120, 113, 113, 99, 113, 74, 141, 272, 255, 244, 233, 226, 212, 212, 187, 215, 141, 272, 527, 499

Offset: 0

Author

Paul D. Hanna, Mar 04 2004

Keywords

• nonn
• tabl

Comments

First column and main diagonal forms A092684. Row sums form A092685.

This triangle is the cascadence of binomial (1+x). 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

Examples

			Rows begin:
1;
1, 1;
2, 1, 2;
3, 3, 2, 3;
6, 5, 5, 3, 6;
11, 10, 8, 9, 6, 11;
21, 18, 17, 15, 17, 11, 21;
39, 35, 32, 32, 28, 32, 21, 39;
74, 67, 64, 60, 60, 53, 60, 39, 74;
141, 131, 124, 120, 113, 113, 99, 113, 74, 141;
272, 255, 244, 233, 226, 212, 212, 187, 215, 141, 272;
527, 499, 477, 459, 438, 424, 399, 402, 356, 413, 272, 527;
1026, 976, 936, 897, 862, 823, 801, 758, 769, 685, 799, 527, 1026; ...
The convolution of each row with {1,1} gives the triangle:
1, 1;
1, 2, 1;
2, 3, 3, 2;
3, 6, 5, 5, 3;
6, 11, 10, 8, 9, 6;
11, 21, 18, 17, 15, 17, 11;
21, 39, 35, 32, 32, 28, 32, 21;
39, 74, 67, 64, 60, 60, 53, 60, 39; ...
which, when flattened, equals the original triangle in flattened form.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1325; rows 0..50 in flattened form.

Crossrefs

Cf. A092684, A092685, A092686, A092689, A120894, A120898.

Programs

PARI

T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), 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 G.F. where F=1+x: */
{T(n,k)=local(A,F=1+x,d=1,G=x,H=1+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

Formula

T(n, k) = T(n-1, k) + T(n-1, k+1) for 0<=k

G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-x))/(1-x) and H(x) is the g.f. of column 0 (A092684). - Paul D. Hanna, Jul 17 2006

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