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

1,5

Comments

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

Links

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

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

Example

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;
...

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 *)

Prog

(PARI) T(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
(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
(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

Crossrefs

Cf. A007318, A077028, A116725.

Sequence in context: A096583 A130154 A208328 * A026615 A026681 A109128

Adjacent sequences: A134395 A134396 A134397 * A134399 A134400 A134401

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 23 2007

Extensions

Extended by Roger L. Bagula, Nov 02 2008

Edited by G. C. Greubel, Nov 29 2019

Status

approved