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A194582
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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.
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1
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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