A172179 (1,[99n+1]) Pascal Triangle.

1, 1, 100, 1, 101, 199, 1, 102, 300, 298, 1, 103, 402, 598, 397, 1, 104, 505, 1000, 995, 496, 1, 105, 609, 1505, 1995, 1491, 595, 1, 106, 714, 2114, 3500, 3486, 2086, 694, 1, 107, 820, 2828, 5614, 6986, 5572, 2780, 793, 1, 108, 927, 3648, 8442, 12600, 12558

Offset: 1

Author

Mark Dols, Jan 28 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1, 100;
  1, 101, 199;
  1, 102, 300,  298;
  1, 103, 402,  598,  397;
  1, 104, 505, 1000,  995,   496;
  1, 105, 609, 1505, 1995,  1491,   595;
  1, 106, 714, 2114, 3500,  3486,  2086,  694;
  1, 107, 820, 2828, 5614,  6986,  5572, 2780,  793;
  1, 108, 927, 3648, 8442, 12600, 12558, 8352, 3573, 892;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000225, A007318, A172171, A172178.

Programs

Mathematica

Table[99*Binomial[n-1, k-2] + Binomial[n-1, k-1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Apr 27 2022 *)

SageMath

flatten([[98*binomial(n-1,k-2) + binomial(n,k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 27 2022

Formula

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 100, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Dec 26 2013

From G. C. Greubel, Apr 27 2022: (Start)

T(n, k) = 99*binomial(n-1, k-2) + binomial(n-1, k-1).

T(n, n) = A172178(n-1).

Sum_{k=1..n} T(n, k) = 100*A000225(n-1) + 1. (End)