A146988 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.

1, 1, 1, 1, 6, 1, 1, 19, 19, 1, 1, 68, 134, 68, 1, 1, 261, 778, 778, 261, 1, 1, 1030, 4111, 6164, 4111, 1030, 1, 1, 4103, 20501, 40995, 40995, 20501, 4103, 1, 1, 16392, 98332, 245816, 327750, 245816, 98332, 16392, 1, 1, 65545, 458788, 1376340, 2293886, 2293886, 1376340, 458788, 65545, 1

Offset: 0

Author

Roger L. Bagula, Nov 04 2008

Keywords

• nonn
• tabl

Comments

Row sums are {1, 2, 8, 40, 272, 2080, 16448, 131200, 1048832, 8389120, 67109888, ...} = {1, 2, 8*A081342(n)}. (modified by G. C. Greubel, Jan 09 2020)

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    6,    1;
  1,   19,   19,    1;
  1,   68,  134,   68,    1;
  1,  261,  778,  778,  261,    1;
  1, 1030, 4111, 6164, 4111, 1030, 1;
		

Links

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

Crossrefs

Cf. A028262, A081342.

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,4) ))); # 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,4): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 09 2020

Maple

q:=4; 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] + 4^(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) + 4^(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, 4) 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 4*(2^n + 8^n) otherwise. - G. C. Greubel, Jan 09 2020

Extensions

Edited by G. C. Greubel, Jan 09 2020