0,4

The g.f. for the row polynomials P(n,x) := sum(a(n,m)*x^m,m=0..n) is 1/(1-(1+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.

The column sequences (without leading zeros) give: A001045(n+1), A073371-9 for m=0..9. Row sums give A002605.

Riordan array (1/(1-x-2x^2),x/(1-x-2x^2)). - Paul Barry (pbarry(AT)wit.ie), Mar 15 2005

W. Lang, First 10 rows.

a(n, m)=sum(binomial(n-k, m)*binomial(n-m-k, k)*2^k, k=0..floor((n-m)/2)) if n>m, else 0.

a(n, m)=(1*(n-m+1)*a(n, m-1)+2*2*(n+m)*a(n-1, m-1))/((1^2+4*2)*m), n>=m>=1, a(n, 0)=A001045(n+1), n>=0, else 0.

a(n, m)= (p(m-1, n-m)*1*(n-m+1)*a(n-m+1)+q(m-1, n-m)*2*(n-m+2)*a(n-m))/(m!*9^m), n>=m>=1, with a(n)=a(n, m=0) := A001045(n+1), else 0; p(k, n) := sum(A(k, l)*n^(k-l), l=0..k) and q(k, n) := sum(B(k, l)*n^(k-l), l=0..k) with the number triangles A(k, m) := A073399(k, m) and B(k, m) := A073400(k, m).

G.f. for column m (without leading zeros): 1/(1-(1+2*x)*x)^(m+1), m>=0.

Number triangle T(n, k) with T(n, 0)=A001045(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+2T(n-2, k)+T(n-1, k) otherwise. - Paul Barry (pbarry(AT)wit.ie), Mar 15 2005

{1},{1,1},{3,2,1},... (lower triangular matrix n>=m>=0).

Sequence in context: A138483 A065366 A092879 this_sequence A129675 A081277 A079628

Adjacent sequences: A073367 A073368 A073369 this_sequence A073371 A073372 A073373

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 2, 2002