A146987 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.

1, 1, 1, 1, 5, 1, 1, 12, 12, 1, 1, 31, 60, 31, 1, 1, 86, 253, 253, 86, 1, 1, 249, 987, 1478, 987, 249, 1, 1, 736, 3666, 7325, 7325, 3666, 736, 1, 1, 2195, 13150, 32861, 43810, 32861, 13150, 2195, 1, 1, 6570, 45963, 137865, 229761, 229761, 137865, 45963, 6570, 1

Offset: 0

Author

Roger L. Bagula, Nov 04 2008

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 7, 26, 124, 680, 3952, 23456, 140224, 840320, 5039872}.

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   5,   1;
  1,  12,  12,    1;
  1,  31,  60,   31,   1;
  1,  86, 253,  253,  86,   1;
  1, 249, 987, 1478, 987, 249, 1;
		

Links

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

Crossrefs

Cf. A028262.

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

Maple

q:=3; 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] + 3^(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) + 3^(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, 3) 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) + 3^(n-1)*binomial(n-2, k -1) otherwise.

Extensions

Edited by G. C. Greubel, Jan 09 2020