A174044 Triangle T(n, k, q) = Sum_{j=0..10} q^j * floor( binomial(n+1,k)*binomial(n-1,k-1)/(2^j*(n+1)) ) for q = 2, read by rows.

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 36, 92, 36, 1, 1, 49, 276, 276, 49, 1, 1, 93, 673, 1265, 673, 93, 1, 1, 124, 1484, 4004, 4004, 1484, 124, 1, 1, 204, 2832, 12400, 18060, 12400, 2832, 204, 1, 1, 237, 5244, 26416, 57580, 57580, 26416, 5244, 237, 1, 1, 289, 7729, 53024, 151756, 211692, 151756, 53024, 7729, 289, 1

Offset

1,5

Comments

Row sums are: {1, 2, 7, 34, 166, 652, 2799, 11226, 48934, 178956, ...}.

Links

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

Formula

T(n, k, q) = Sum_{j=0..10} q^j * floor( binomial(n+1,k)*binomial(n-1,k-1)/(2^j*(n+1)) ) for q = 2.

Example

The triangle begins as:
  1;
  1,   1;
  1,   5,    1;
  1,  16,   16,     1;
  1,  36,   92,    36,     1;
  1,  49,  276,   276,    49,     1;
  1,  93,  673,  1265,   673,    93,     1;
  1, 124, 1484,  4004,  4004,  1484,   124,    1;
  1, 204, 2832, 12400, 18060, 12400,  2832,  204,   1;
  1, 237, 5244, 26416, 57580, 57580, 26416, 5244, 237, 1;

Mathematica

T[n_, k_, q_]:= Sum[q^j*Floor[Binomial[n-1, k-1]*Binomial[n+1, k]/(2^j*(n+1))], {j, 0, 10}];
Table[T[n, k, 2], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 16 2021 *)

Prog

(Magma)
T:= func< n, k, q | (&+[ q^j*Floor( Binomial(n+1, k)*Binomial(n-1, k-1)/(2^j*(n+1)) ): j in [0..10]]) >;
[T(n, k, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2021
(Sage)
def T(n, k, q): return sum( q^j*( (binomial(n+1, k)*binomial(n-1, k-1)//(2^j*(n+1))) ) for j in (0..10))
flatten([[T(n, k, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 16 2021

Crossrefs

Cf. A174043 (q=1), this sequence (q=2), A174045 (q=3).

Sequence in context: A232103 A292357 A156920 * A174159 A074060 A157637

Adjacent sequences: A174041 A174042 A174043 * A174045 A174046 A174047

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Mar 06 2010

Extensions

Edited by G. C. Greubel, Apr 16 2021

Status

approved