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A273914 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.

%I A273914 #14 Aug 20 2018 11:22:29
%S A273914 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 A273914 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 A273914 45,120,210,245,192,106,41,10,1,1,11,55,165,330,453,430,290,143,50,11,1
%N A273914 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 A273914 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 A273914 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 A273914 Antidiagonal sums is A062257.
%F A273914 A(n, 3) = A000292(n). A(n, 4) = A000332(n).
%e A273914 Array begins:
%e A273914 n\m  0  1   2   3    4    5
%e A273914 --+------------------------
%e A273914 0 |  1  1   1   1    1    1
%e A273914 1 |  1  2   3   4    5    6
%e A273914 2 |  1  3   6  10   15   20
%e A273914 3 |  1  4  10  20   35   53
%e A273914 4 |  1  5  15  35   70  121
%e A273914 5 |  1  6  21  56  126  245
%t A273914 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 A273914 Table[A[n-m, m], {n, 0, 11}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 20 2018, from PARI *)
%o A273914 (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 A273914 Cf. A000292, A000332, A062257.
%Y A273914 Main diagonal gives A275046.
%K A273914 nonn,tabl
%O A273914 0,5
%A A273914 _Michael Somos_, Jun 03 2016