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Array A by antidiagonals going up: A(n, m) is the number of (0, 1)-strings with n 0's and m 1's that do not contain 10101101 or 1110101 as substrings.
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%I #14 Aug 20 2018 11:22:29

%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,20,15,6,1,1,7,21,

%T 35,35,20,7,1,1,8,28,56,70,53,26,8,1,1,9,36,84,126,121,76,33,9,1,1,10,

%U 45,120,210,245,192,106,41,10,1,1,11,55,165,330,453,430,290,143,50,11,1

%N Array A by antidiagonals going up: A(n, m) is the number of (0, 1)-strings with n 0's and m 1's that do not contain 10101101 or 1110101 as substrings.

%H R. Pemantle and M. C. Wilson, <a href="http://dx.doi.org/10.1137/050643866">Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions</a>, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 255, equ. (5.2)

%F G.f.: (1 + x^2*y^3 + x^2*y^4 + x^3*y^4 - x^3*y^6) / (1 - x - y + x^2*y^3 - x^3*y^3 - x^4*y^4 - x^3*y^6 + x^4*y^6).

%F Antidiagonal sums is A062257.

%F A(n, 3) = A000292(n). A(n, 4) = A000332(n).

%e Array begins:

%e n\m 0 1 2 3 4 5

%e --+------------------------

%e 0 | 1 1 1 1 1 1

%e 1 | 1 2 3 4 5 6

%e 2 | 1 3 6 10 15 20

%e 3 | 1 4 10 20 35 53

%e 4 | 1 5 15 35 70 121

%e 5 | 1 6 21 56 126 245

%t A[n_, m_] := If[n<0 || m<0, 0, SeriesCoefficient[ SeriesCoefficient[(1 + x^2*y^3 + x^2*y^4 + x^3*y^4 - x^3*y^6)/(1 -x - y + x^2*y^3 - x^3*y^3 - x^4*y^4 - x^3*y^6 + x^4*y^6), {x, 0, n}], {y, 0, m}]];

%t Table[A[n-m, m], {n, 0, 11}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 20 2018, from PARI *)

%o (PARI) {A(n, m) = if( n<0 || m<0, 0, polcoeff( polcoeff( (1 + x^2*y^3 + x^2*y^4 + x^3*y^4 - x^3*y^6) / (1 - x - y + x^2*y^3 - x^3*y^3 - x^4*y^4 - x^3*y^6 + x^4*y^6) + x * O(x^n), n) + y * O(y^m), m))};

%Y Cf. A000292, A000332, A062257.

%Y Main diagonal gives A275046.

%K nonn,tabl

%O 0,5

%A _Michael Somos_, Jun 03 2016