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

0,4

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

Links

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

Formula

T(n, k) = 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)=1.

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A033791 A039913 A108617 * A172089 A057475 A024376

Adjacent sequences: A092680 A092681 A092682 * A092684 A092685 A092686

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 04 2004

Status

approved