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.
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, 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, 45, 120, 210, 245, 192, 106, 41, 10, 1, 1, 11, 55, 165, 330, 453, 430, 290, 143, 50, 11, 1
Offset: 0
Examples
Array begins: n\m 0 1 2 3 4 5 --+------------------------ 0 | 1 1 1 1 1 1 1 | 1 2 3 4 5 6 2 | 1 3 6 10 15 20 3 | 1 4 10 20 35 53 4 | 1 5 15 35 70 121 5 | 1 6 21 56 126 245
Links
- R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 255, equ. (5.2)
Programs
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Mathematica
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}]]; Table[A[n-m, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 20 2018, from PARI *) -
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))};