login
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.
13
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
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 04 2004
STATUS
approved