A146986 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 12, 22, 12, 1, 1, 21, 58, 58, 21, 1, 1, 38, 143, 212, 143, 38, 1, 1, 71, 341, 675, 675, 341, 71, 1, 1, 136, 796, 1976, 2630, 1976, 796, 136, 1, 1, 265, 1828, 5460, 9086, 9086, 5460, 1828, 265, 1, 1, 522, 4141, 14456, 28882, 36092, 28882, 14456, 4141, 522, 1

Offset: 0

Author

Roger L. Bagula, Nov 04 2008

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 6, 16, 48, 160, 576, 2176, 8448, 33280, 132096, ...} = {1, 2, 2*A242985(n)}. (modified by G. C. Greubel, Jan 09 2020)

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1,  7,   7,   1;
  1, 12,  22,  12,   1;
  1, 21,  58,  58,  21,  1;
  1, 38, 143, 212, 143, 38, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A028262, A242985.

Programs

GAP

T:= function(n,k,q)
    if n<2 then return Binomial(n,k);
    else return Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1);
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k,2) ))); # G. C. Greubel, Jan 09 2020

Magma

T:= func< n,k,q | n lt 2 select Binomial(n,k) else Binomial(n,k) + q^(n-1)*Binomial(n-2,k-1) >;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020

Maple

q:=2; seq(seq( `if`(n<2, binomial(n,k), binomial(n,k) + q^(n-1)*binomial(n-2,k-1)), k=0..n), n=0..10); # G. C. Greubel, Jan 09 2020

Mathematica

Table[If[n<2, Binomial[n, m], Binomial[n, m] + 2^(n-1)*Binomial[n-2, m-1]], {n, 0, 10}, {m, 0, n}]//Flatten

PARI

T(n,k) = if(n<2, binomial(n,k), binomial(n,k) + 2^(n-1)*binomial(n-2,k-1) ); \\ G. C. Greubel, Jan 09 2020

Sage

@CachedFunction
def T(n, k, q):
    if (n<2): return binomial(n,k)
    else: return binomial(n,k) + q^(n-1)*binomial(n-2,k-1)
[[T(n, k, 2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 09 2020

Formula

T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 2^(n-1) * binomial(n-2, k-1) otherwise.

Sum_{k=0..n} T(n,k) = n+1 for n < 2 and 16*binomial(2^(n-3) + 1, 2) otherwise. - G. C. Greubel, Jan 09 2020

Extensions

Edited by G. C. Greubel, Jan 09 2020