0,5

Row coefficients are the absolute values of the coefficients of the characteristic polynomials of the n X n matrices A(n) with A(n)_i,i=2, i>0, A(n)_i,j=1, otherwise (starts with (0,0) position).

Number triangle T(n, k)=if(k<=n, if(k=0, 1, binomial(n-k, k-1)((k+1)(n-k)+k)/k, 0), 0); T(n, 0)=1, T(0, k)=0, k>0, T(n, k)=T(n-1, k-1)+T(n-1, k)+binomial(n-2, k-1); Column k is generated by (1+kx)x^k/(1-x)^(k+1); rows are coefficients of the polynomials P(0, x)=1, P(n, x)=(1+x)^(n-2)(1+(n+1)x+x^2), n>0

T(n,k)=sum{j=0..n, C(k,k-j)*C(n-k,j)*(j+1)}*[k<=n]; - Paul Barry (pbarry(AT)wit.ie), Oct 28 2006

A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1; p(x, 1) =(x - 1); p(x, 2) = (1 - 3 x + x^2); p(x,n)=(-1 + x)^(n - 2)*(1 - (n + 1)*x + x^2); t(n,m)=Coefficients(p(x,n)). - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 21 2008

Comment from Roger L. Bagula, Oct 21 2008: The triangle begins:

{1},

{1, 1},

{1, 3, 1},

{1, 5, 5, 1},

{1, 7, 12, 7, 1},

{1, 9, 22, 22, 9, 1},

{1, 11, 35, 50, 35, 11, 1},

{1, 13, 51, 95, 95, 51, 13, 1},

{1, 15, 70, 161, 210, 161, 70, 15, 1},

{1, 17, 92, 252, 406, 406, 252, 92, 17, 1},

{1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1}

...

Clear[p, x, n, m] p[x, 0] = 1; p[x, 1] = (x - 1); p[x, 2] = (1 - 3 x + x^2); p[x, 3] = (-1 + x)* (1 - 4 x + x^2); p[x, 4] = (-1 + x)^2 *(1 - 5 x + x^2); p[x_, n_] := p[x, n] = (-1 + x)^(n - 2)*(1 - (n + 1)*x + x^2); Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%] - Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 21 2008

Row sums are A045623. Columns include A005408, A000326, A002412, A002418.

Sequence in context: A026681 A109128 A113245 this_sequence A128254 A026714 A008288

Adjacent sequences: A103447 A103448 A103449 this_sequence A103451 A103452 A103453

easy,nonn,tabl

Paul Barry (pbarry(AT)wit.ie), Feb 06 2005