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A103450 A figurate number triangle read by rows. 6
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, 1, 21, 145, 525, 1170, 1722 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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).

The triangle can be generated by the matrix multiplication A007318 * A114219s, where

A114219s = 0; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,3,1; 0,-1,2,-3,4,1;...= A097807 * A128229 is a signed variant of A114219. - Gary W. Adamson, Feb 20 2007

REFERENCES

G. Chiaselotti, W. Keith, P. A, Oliverio, Two Self-Dual Lattices of Signed Integer Partitions, Appl. Math. Inf. Sci. 8, No. 6, 3191-3199 (2014).

LINKS

Table of n, a(n) for n=0..71.

FORMULA

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, 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, Oct 21 2008

T(2n,n) = A141222(n). - Emanuele Munarini, Jun 01 2012

Conjecture: g.f. is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). [Joerg Arndt, Aug 27 2013]

EXAMPLE

From Roger L. Bagula, Oct 21 2008: (Start)

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;

... (End)

MATHEMATICA

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, Oct 21 2008 *)

CROSSREFS

Row sums are A045623.

Columns include A005408, A000326, A002412, A002418.

Sequence in context: A026681 A109128 A113245 * A128254 A277930 A026714

Adjacent sequences:  A103447 A103448 A103449 * A103451 A103452 A103453

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 06 2005

STATUS

approved

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)