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

0,5

Links

Table of n, a(n) for n=0..77.

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)

Formula

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).

Antidiagonal sums is A062257.

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A000292, A000332, A062257.

Main diagonal gives A275046.

Sequence in context: A144398 A034932 A180183 * A094495 A154926 A117440

Adjacent sequences: A273911 A273912 A273913 * A273915 A273916 A273917

Keyword

nonn,tabl

Author

Michael Somos, Jun 03 2016

Status

approved