A103450 A figurate number triangle read by rows.

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, 1722, 1170, 525, 145, 21, 1

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

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

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

Formula

T(n, k) = binomial(n-1, k-1)*(k*(n-k) + n)/k with T(n, 0) = 1.

T(n, k) = T(n-1, k-1) + T(n-1, k) + binomial(n-2, k-1) with T(n, 0) = 1.

Column k is generated by (1+k*x)*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) for n>0.

T(n,k) = Sum_{j=0..n} binomial(k, k-j)*binomial(n-k, j)*(j+1). - Paul Barry, Oct 28 2006

A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1, p(x, 1) = (-1 +x), p(x, 2) = (1 -3*x +x^2), p(x,n) = (-1 +x)^(n-2)*(1 - (n + 1)*x + x^2); T(n, k) = (-1)^(n+k)*coefficient of x^k of ( p(x,n) ). - Roger L. Bagula and Gary W. Adamson, Oct 21 2008

T(2*n+1, n) = A141222(n). - Emanuele Munarini, Jun 01 2012 [corrected by Werner Schulte, Nov 27 2021]

G.f.: is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). - Joerg Arndt, Aug 27 2013

Sum_{k=0..floor(n/2)} T(n-k, k) = (1/5)*((-n+5)*Fibonacci(n+1) + (3*n- 2)*Fibonacci(n)) = A208354(n). - G. C. Greubel, Jun 17 2021

T(2*n, n) = A000984(n) * (n + 2) / 2 for n >= 0. - Werner Schulte, Nov 27 2021

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

(* First program *)
p[x_, n_]:= p[x, n]= If[n==0, 1, (-1+x)^(n-2)*(1 -(n+1)*x +x^2)];
T[n_, k_]:= T[n, k]= (-1)^(n+k)*SeriesCoefficient[p[x, n], {x, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 21 2008 *)(* corrected by G. C. Greubel, Jun 17 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Binomial[n, k]*(n*(k+1) -k^2)/n];
Table[T[n, k], {n, 0, 16}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 17 2021 *)

Prog

(Magma)
A103450:= func< n, k | k eq 0 select 1 else Binomial(n, k)*(k*(n-k) + n)/n >;
[A103450(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 17 2021
(Sage)
def A103450(n, k): return 1 if (k==0) else binomial(n, k)*(k*(n-k) + n)/n
flatten([[A103450(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 17 2021

Crossrefs

Row sums are A045623.

Columns include: A000326, A002412, A002418, A005408.

Cf. A000045, A208354.

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