A134398 Triangle read by rows: T(n, k) = (k-1)*(n-k) + binomial(n-1,k-1).

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 10, 7, 1, 1, 9, 16, 16, 9, 1, 1, 11, 23, 29, 23, 11, 1, 1, 13, 31, 47, 47, 31, 13, 1, 1, 15, 40, 71, 86, 71, 40, 15, 1, 1, 17, 50, 102, 146, 146, 102, 50, 17, 1, 1, 19, 61, 141, 234, 277, 234, 141, 61, 19, 1

Offset: 1

Author

Gary W. Adamson, Oct 23 2007

Keywords

• nonn
• tabl

Comments

Row sums = A116725: (1, 2, 5, 12, 26, 52, ...).

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  5,  1;
  1,  7, 10,  7,  1;
  1,  9, 16, 16,  9,  1;
  1, 11, 23, 29, 23, 11,  1;
  1, 13, 31, 47, 47, 31, 13,  1;
  1, 15, 40, 71, 86, 71, 40, 15, 1;
...
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A007318, A077028, A116725.

Programs

GAP

Flat(List([1..10], n-> List([1..n], k-> (k-1)*(n-k) + Binomial(n-1,k-1) ))); # G. C. Greubel, Nov 29 2019

Magma

[(k-1)*(n-k) + Binomial(n-1,k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 29 2019

Maple

seq(seq( (k-1)*(n-k) + binomial(n-1,k-1), k=1..n), n=1..10); # G. C. Greubel, Nov 29 2019

Mathematica

p[x_, n_]:= p[x,n]= If[n==0, 1, (x+1)^n +Sum[(n-m)*m*x^m*(1 +x^(n-2*m)), {m, 1, n- 1}]/2]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 02 2008 *)
Table[(k-1)*(n-k) + Binomial[n-1, k-1], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 29 2019 *)

PARI

T(n,k) = (k-1)*(n-k) + binomial(n-1,k-1); \\ G. C. Greubel, Nov 29 2019

Sage

[[(k-1)*(n-k) + binomial(n-1,k-1) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 29 2019

Formula

T(n, k) = A077028(n,k) + A007318(n,k) - 1.

Let p(x, n) = (1+x)^n + (1/2) * Sum_{j=1..n-1} (n-j)*j*x^j*(1 + x^(n - 2*j)) with p(x, 0) = 1, then T(n, k) = Coefficients(p(x,n)). - Roger L. Bagula, Nov 02 2008

T(n, k) = (k-1)*(n-k) + binomial(n-1,k-1). - G. C. Greubel, Nov 29 2019

Extensions

Extended by Roger L. Bagula, Nov 02 2008

Edited by G. C. Greubel, Nov 29 2019