A275046 -id:A275046

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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.

+10
1

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

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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.

KEYWORD

nonn,tabl

AUTHOR

Michael Somos, Jun 03 2016

STATUS

approved

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