A137153 Triangle, read by rows, where T(n,k) = C(2^k + n-k-1, n-k).

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 8, 1, 1, 5, 20, 36, 16, 1, 1, 6, 35, 120, 136, 32, 1, 1, 7, 56, 330, 816, 528, 64, 1, 1, 8, 84, 792, 3876, 5984, 2080, 128, 1, 1, 9, 120, 1716, 15504, 52360, 45760, 8256, 256, 1, 1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896

Offset

0,5

Comments

Matrix inverse is A137156.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1080; rows 0..45 of flattened triangle.

Example

Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   4,    1;
  1,  4,  10,    8,     1;
  1,  5,  20,   36,    16,      1;
  1,  6,  35,  120,   136,     32,      1;
  1,  7,  56,  330,   816,    528,     64,      1;
  1,  8,  84,  792,  3876,   5984,   2080,    128,     1;
  1,  9, 120, 1716, 15504,  52360,  45760,   8256,   256,   1;
  1, 10, 165, 3432, 54264, 376992, 766480, 357760, 32896, 512, 1;
  ...

Mathematica

Table[Binomial[2^k+n-k-1, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Mar 06 2017 *)

Prog

(PARI) {T(n, k)=binomial(2^k+n-k-1, n-k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = polcoeff(1/(1-x+x*O(x^n))^(2^k), n-k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A137154 (row sums), A137155 (antidiagonal sums), A060690 (central terms); A137156 (matrix inverse).

Cf. A092056 (same with reflected rows).

Sequence in context: A162717 A122175 A073165 * A340814 A063841 A256161

Adjacent sequences: A137150 A137151 A137152 * A137154 A137155 A137156

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 24 2008

Status

approved