A124959 Triangle read by rows: T(n,k) = a(k)*binomial(n,k) (0 <= k <= n), where a(0)=1, a(1)=2, a(k) = a(k-1) + 3*a(k-2) for k >= 2 (a(k) = A006138(k)).

1, 1, 2, 1, 4, 5, 1, 6, 15, 11, 1, 8, 30, 44, 26, 1, 10, 50, 110, 130, 59, 1, 12, 75, 220, 390, 354, 137, 1, 14, 105, 385, 910, 1239, 959, 314, 1, 16, 140, 616, 1820, 3304, 3836, 2512, 725, 1, 18, 180, 924, 3276, 7434, 11508, 11304, 6525, 1667, 1, 20, 225, 1320, 5460, 14868, 28770, 37680, 32625, 16670, 3842

Offset: 0

Author

Gary W. Adamson, Nov 13 2006

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = A006190(n+1).

Examples

			First few rows of the triangle:
  1;
  1,   2;
  1,   4,   5;
  1,   6,  15,  11;
  1,   8,  30,  44,  26;
  1,  10,  50, 110, 130,  59;
  ...
		

Links

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

Crossrefs

Cf. A006138, A006190.

Programs

Magma

function b(k)
  if k lt 2 then return k+1;
  else return b(k-1) + 3*b(k-2);
  end if;
  return b;
end function;
[Binomial(n,k)*b(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2019

Maple

a:=proc(n) if n=0 then 1 elif n=1 then 2 else a(n-1)+3*a(n-2) fi end: T:=(n,k)->a(k)*binomial(n,k): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_]:= T[n, k]= Simplify[(I*Sqrt[3])^(k-1)*Binomial[n,k]*(I*Sqrt[3]* ChebyshevU[k, 1/(2*I*Sqrt[3])] + ChebyshevU[k-1, 1/(2*I*Sqrt[3])])];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

PARI

b(k) = if(k<2, k+1, b(k-1) + 3*b(k-2));
T(n,k) = binomial(n,k)*b(k);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 19 2019

Sage

@CachedFunction
def b(k):
    if (k<2): return k+1
    else: return b(k-1) + 3*b(k-2)
[[binomial(n, k)*b(k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 19 2019

Extensions

Edited by N. J. A. Sloane, Dec 03 2006