A046918 Triangle of coefficients of polynomials p(n), with p(3)=1, p(n) = (1 - t^(2*n - 4))*(1 - t^(2*n - 3))*p(n - 1)/((1 - t^(n - 3))*(1 - t^n)).
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 2, 3, 4, 5, 7, 7, 8, 8, 8, 7, 7, 5, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 16, 19, 20, 23, 23, 24, 23, 23, 20, 19, 16, 14, 11, 9, 6, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 28, 33, 40, 45, 52, 57, 63, 66, 70, 71
Offset: 3
Examples
1; 1 + t + t^2 + t^3 + t^4 + t^5; t^10 + t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 3*t^5 + 3*t^4 + 2*t^3 + 2*t^2 + t + 1; ...
Links
- J. Riordan, The number of score sequences in tournaments, J. Combin. Theory, 5 (1968), 87-89. [But beware errors.]
Crossrefs
Cf. A046919.
Programs
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Maple
p := proc(n) option remember; if n = 3 then 1 else (1-t^(2*n-4))*(1-t^(2*n-3))*p(n-1)/((1-t^(n-3))*(1-t^n)); fi; end;
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Mathematica
p[3] = 1; p[n_] := p[n] = (1 - t^(2*n-4))*(1 - t^(2*n-3))*p[n-1]/((1 - t^(n-3))*(1 - t^n)) // Simplify; Table[ CoefficientList[ Series[p[n], {t, 0, n^3}], t], {n, 3, 8}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)
Extensions
Keyword tabf by Michel Marcus, Dec 05 2014