A194582 Triangle T(n,k), read by rows, given by (0, 3, -7/3, -2/21, 3/7, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 3, 1, 0, 2, 6, 1, 0, 2, 13, 9, 1, 0, 2, 16, 33, 12, 1, 0, 2, 20, 69, 62, 15, 1, 0, 2, 24, 108, 188, 100, 18, 1, 0, 2, 28, 156, 401, 400, 147, 21, 1, 0, 2, 32, 212, 704, 1115, 732, 203, 24, 1, 0, 2, 36, 276, 1120, 2433, 2547, 1211, 268, 27, 1

Offset: 0

Author

Philippe Deléham, Jan 23 2012

Keywords

• nonn
• tabl

Comments

Riordan array (1, x*(1+2x-x^2)/(1-x)).

Row sums are (Fibonacci(n+1))^2 = A007598(n+1).

T(n, k) is the number of ordered pairs of Fibonacci bit strings of length n with the number of matching 1 bits in the same position is k. A Fibonacci bit string begins a 1 bit and no two consecutive bits are 0 bits. - Michael Somos, Feb 28 2020

Examples

			Triangle begins:
  1;
  0,   1;
  0,   3,   1;
  0,   2,   6,   1;
  0,   2,  13,   9,   1;
  0,   2,  16,  33,  12,   1;
  0,   2,  20,  69,  62,  15,   1;
  0,   2,  24, 108, 188, 100,  18,   1;
  0,   2,  28, 156, 401, 400, 147,  21,   1;
T(3, 2) = 6 enumerates the pairs of Fibonacci bit string of length 3 with 2 matching 1 bits: (101, 101), (101, 111), (110, 110), (110, 111), (111, 101), (111, 110). - _Michael Somos_, Feb 28 2020
		

Links

• Indranil Ghosh, Rows 0..100, flattened

Crossrefs

Cf. A000045, A007598. Diagonals: A000012, A008585, A062708.

Programs

Mathematica

nmax=10; Flatten[CoefficientList[Series[CoefficientList[Series[(1 - x)/(1 - x - x*y - 2*x^2*y + x^3*y)  , {x,  0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017, after  R. J. Mathar *)

PARI

T(n,k) = if(n==k, 1, if(k==0, 0, if(n>1 && k==n - 1, 3*k, T(n - 1, k) + T(n - 1,k - 1) + 2*T(n - 2,k - 1) - T(n-3,k-1))));
{for(n=0, 10, for(k=0, n, print1(T(n,k),", ");); print();); } \\ Indranil Ghosh, Mar 10 2017

Formula

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1) - T(n-3,k-1).

G.f.: (1-x)/(1-x-x*y-2*x^2*y+x^3*y). - R. J. Mathar, Aug 11 2015