A194582 - OEIS

%I #42 Feb 28 2020 13:25:00

%S 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,

%T 24,108,188,100,18,1,0,2,28,156,401,400,147,21,1,0,2,32,212,704,1115,

%U 732,203,24,1,0,2,36,276,1120,2433,2547,1211,268,27,1

%N 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.

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

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

%C 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

%H Indranil Ghosh, <a href="/A194582/b194582.txt">Rows 0..100, flattened</a>

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

%F G.f.: (1-x)/(1-x-x*y-2*x^2*y+x^3*y). - _R. J. Mathar_, Aug 11 2015

%e Triangle begins:

%e   1;

%e   0,   1;

%e   0,   3,   1;

%e   0,   2,   6,   1;

%e   0,   2,  13,   9,   1;

%e   0,   2,  16,  33,  12,   1;

%e   0,   2,  20,  69,  62,  15,   1;

%e   0,   2,  24, 108, 188, 100,  18,   1;

%e   0,   2,  28, 156, 401, 400, 147,  21,   1;

%e 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

%t 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_ *)

%o (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))));

%o {for(n=0, 10, for(k=0, n, print1(T(n,k),", ");); print();); } \\ _Indranil Ghosh_, Mar 10 2017

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

%K nonn,tabl

%O 0,5

%A _Philippe Deléham_, Jan 23 2012