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

1,3

Links

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

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)

Example

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;

Mathematica

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

Prog

(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

Crossrefs

Cf. A000225, A007318, A172171, A172178.

Sequence in context: A267440 A154230 A173183 * A147804 A164847 A273479

Adjacent sequences: A172176 A172177 A172178 * A172180 A172181 A172182

Keyword

nonn,tabl

Author

Mark Dols, Jan 28 2010

Status

approved