A135936 - OEIS

%I #26 Mar 11 2023 04:52:24

%S 1,1,1,2,1,1,1,0,-2,1,-1,-3,1,-2,-3,2,1,-3,-2,5,1,-4,0,8,-2,1,-5,3,10,

%T -7,1,-6,7,10,-15,2,1,-7,12,7,-25,9,1,-8,18,0,-35,24,-2,1,-9,25,-12,

%U -42,49,-11,1,-10,33,-30,-42,84,-35,2,1,-11,42,-55,-30,126,-84,13,1,-12,52,-88,0,168,-168,48,-2,1,-13,63,-130,55,198,-294

%N Irregular triangle read by rows: row n gives coefficients of Boubaker polynomial B_n(x) in order of decreasing exponents (another version).

%C See A135929 and A138034 for further information.

%H R. J. Mathar, Mar 11 2008, <a href="/A135936/b135936.txt">Table of n, a(n) for n = 0..160</a>

%F Conjectures from _Thomas Baruchel_, Jun 03 2018: (Start)

%F T(n,m) = 4*A115139(n+1,m) - 3*A132460(n,m).

%F T(n,m) = (-1)^m * (binomial(n-m, m) - 3*binomial(n-m-1, m-1)). (End)

%e The Boubaker polynomials B_0(x), B_1(x), B_2(x), ... are:

%e   1

%e   x

%e   x^2    + 2

%e   x^3    + x

%e   x^4             - 2

%e   x^5    - x^3  - 3*x

%e   x^6  - 2*x^4  - 3*x^2    + 2

%e   x^7  - 3*x^5  - 2*x^3  + 5*x

%e   x^8  - 4*x^6           + 8*x^2    - 2

%e   x^9  - 5*x^7  + 3*x^5 + 10*x^3  - 7*x

%e   ...

%p A135936 := proc(n,m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2),t=0,n), x=0,m) ; end: for n from 0 to 25 do for m from n to 0 by -2 do printf("%d, ",A135936(n,m)) ; od; od; # _R. J. Mathar_, Mar 11 2008

%t T[n_, m_] := SeriesCoefficient[SeriesCoefficient[

%t    (1+3*t^2)/(1-x*t+t^2), {t, 0, n}], {x, 0, m}];

%t Table[T[n, m], {n, 0, 25}, {m, n, 0, -2}] // Flatten (* _Jean-François Alcover_, Mar 11 2023, after _R. J. Mathar_ *)

%Y Cf. A138034.

%K sign,tabf

%O 0,4

%A _N. J. A. Sloane_, Mar 09 2008

%E More terms from _R. J. Mathar_, Mar 11 2008