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 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 (list; table; graph; refs; listen; history; text; internal format)
 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 FORMULA T(n, k) = T(n-1, k) + T(n-1, k+1) for 0<=kn, 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

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Last modified December 14 17:32 EST 2019. Contains 329979 sequences. (Running on oeis4.)