1,3

The Boubaker Polynomials A137276 are differentiated once and the coefficients listed in order of increasing exponents.

Row sums are 1, 2, 4, 4, -1, -8, -9, 0, 12, 14, 1.. with g.f. x*(1+3*x^2)/(x^2-x+1)^2.

The integrals Int_{x=-1..1} P(n,x)*P(m,x)/sqrt(1-x^2) dx are zero if n+m is odd, else rational multiples of Pi, where P(n,x) = (d/dx) B_n(x).

Karem Boubaker, On modified Boubaker polynomials..., Trends in Appl. Sci. Research, 2 (2007), 540-544.

Karem Boubaker et al., Enhancement of pyrolysis spray disposal performance ..., Eur. Phys. J. Appl. Phys., 37 (2007), 105-109.

Hedi Labiadh and Karem Boubaker, A Sturm-Liouville shaped characteristic differential equation ..., Differential Equations and Control Processes, No. 2 (2007).

a(n,k) = (k+1)*A137276(n,k+1) .

The coefficients and differentiated polynomials start in row n=1, columns 0<=k<n, as:

{1}, = 1

{0, 2}, = 2x

{1, 0, 3}, = 1+3x^2

{0, 0, 0, 4}, = 4x^3

{-3, 0, -3, 0, 5}, = -3-3x^2+5x^4

{0, -6, 0, -8, 0, 6},

{5, 0, -6, 0, -15, 0, 7},

{0, 16, 0, 0, 0, -24, 0, 8},

{-7, 0, 30, 0, 15, 0, -35, 0, 9},

{0, -30, 0, 40, 0,42, 0, -48, 0, 10},

{9, 0, -75, 0, 35, 0, 84, 0, -63, 0, 11}

B := proc(n, x) if n = 0 then 1; else add( (-1)^j*binomial(n-j, j)*(n-4*j)/(n-j)*x^(n-2*j), j=0..n/2) ; fi; end:

A136255 := proc(n, k) diff( B(n, x), x) ; coeftayl(%, x=0, k) ; end: seq( seq(A136255(n, k), k=0..n-1), n=1..15) ;

Clear[B, x, n] B[x, 0] = 1; B[x, 1] = x; B[x, 2] = 2 + x^2; B[x, 3] = x + x^3; B[x, 4] = -2 + x^4; B[x_, n_] := B[x, n] = x*B[x, n - 1] - B[x, n - 2] P[x_, n_] := D[B[x, n + 1], x] Table[ExpandAll[P[x, n]], {n, 0, 10}] a = Table[CoefficientList[P[x, n], x], {n, 0, 10}] Flatten[a]

Cf. A138034, A135929, A135936, A137276, A137277, A137289.

Sequence in context: A145490 A070677 A029584 this_sequence A159813 A157409 A165252

Adjacent sequences: A136252 A136253 A136254 this_sequence A136256 A136257 A136258

tabl,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 17 2008

Edited by the Associate Editors of the OEIS, Aug 27 2009