A337129 Triangular array read by rows: T(n,0) = 2^n, T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j) for k > 0.

1, 2, 3, 4, 6, 16, 8, 12, 32, 84, 16, 24, 64, 168, 440, 32, 48, 128, 336, 880, 2304, 64, 96, 256, 672, 1760, 4608, 12064, 128, 192, 512, 1344, 3520, 9216, 24128, 63168, 256, 384, 1024, 2688, 7040, 18432, 48256, 126336, 330752, 512, 768, 2048, 5376, 14080, 36864, 96512, 252672, 661504, 1731840

Offset: 0

Author

Oboifeng Dira, Sep 14 2020

Keywords

• nonn
• tabl

Examples

			The triangle  T(n,k) begins:
   n\k  0    1    2    3    4    5
   0:   1
   1:   2    3
   2:   4    6    16
   3:   8    12   32  84
   4:   16   24   64  168  440
   5:   32   48   128 336  880  2304
   ...
T(3,2) = ((3+sqrt(5))^3-(3-sqrt(5))^3)*(2)/(4*sqrt(5)) = (64*sqrt(5))/(2*sqrt(5)) = 32.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Cf. A000079 (1st column), A069429 (diagonal), A018903 (row sums), A001906, A004171.

Programs

Maple

T := proc (n, k) if k = 0 and 0 <= n then 2^n elif 1 <= k and k <= n then round((((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

Mathematica

T[n_, 0] := 2^n;
T[n_, n_] := 2^(n-1) Fibonacci[2n+2];
T[n_, k_] /; 0Jean-François Alcover, Nov 13 2020 *)

PARI

T(n,k) = if (k == 0, 2^n, my(w=quadgen(5, 'w)); ((2*w+2)^(k+1)-(4-2*w)^(k+1))*(2^(n-k))/(4*(2*w-1))); \\ Michel Marcus, Sep 14 2020

PARI

Row(n)={Vecrev(polcoef((1-x*y)*(1-2*x*y)/((1-6*x*y+4*x^2*y^2)*(1-2*x)) + O(x*x^n), n))} \\ Andrew Howroyd, Sep 23 2020

Formula

T(n,0) = 2^n.

T(n,k) = ((3+sqrt(5))^(k+1)-(3-sqrt(5))^(k+1))*(2^(n-k))/(4*sqrt(5)) for 1<=k<=n.

T(n+1,n) = 2*T(n,n).

T(n+m,n) = 2^m*T(n,n), for m>=1.

T(n,n) = A069429(n) = 2^(n-1)*A001906(n+1) for n>=1.

T(2*n,n) = (1/2)*A099157(n+1) = A004171(n-1)*A001906(n+1) for n>=1.

G.f.: (1 - x*y)*(1 - 2*x*y)/((1 - 6*x*y + 4*x^2*y^2)*(1 - 2*x)). - Andrew Howroyd, Sep 23 2020