Real-rooted atlas

A116414 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(52\,n - 32\,n x - 16\,n x^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(118 - 92\,x - 88\,x^{2} - 16\,x^{3} + 198\,n - 60\,n x - 88\,n x^{2} - 16\,n x^{3} + 120\,n^{2} + 48\,n^{2} x + 40\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-39 - 268\,x - 384\,x^{2} - 144\,x^{3} - 16\,x^{4} + 52\,n + 80\,n x + 16\,n x^{2} + 52\,n^{2} + 80\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1729
24228474
3233258848583
4122919481044224164461
5626413064999235045603233416
63115381012815164064010560134464246289
71521194753345970603909201439202966431361281792281
8732306267694040472203306240159536046457679296716825612909362
9348558514607510258836402545608015309840583228814031362042881612851292199007
10164375577774252015821676018247632013206648062633088196788484030464511488358401024653830389
117692254440544878493281864012366244801049549280598589376233864064624606721113216012518407884820484608742736
12357637969207936861653400356888009978080781594632052257358082453394048813911040188916480297932803007488172032409632317900945
131653537271105149335462983024792849990182736552135053604243868899223346447360929998848026786073605495142407769907271147523727368192225600848025
1476083339785253714943616322917670830244558156837337911811232487283744020540739744095827522560331167091208422824960154766233619829555216613376802816163841568610041786
Technical details
Node id
n-2bf63854fb71d0e4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4df8da115d060a29

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       7,        2;
  2:      42,       28,        4;
  3:     233,      258,       84,        8;
  4:    1229,     1948,     1044,      224,       16;
  5:    6264,    13064,     9992,     3504,      560,      32;
  6:   31153,    81012,    81516,    40640,    10560,    1344,      64;
  7:  152119,   475334,   597060,   390920,   143920,   29664,    3136,    128;
  8:  732306,  2676940,  4047220,  3306240,  1595360,  464576,   79296,   7168,   256;
  9: 3485585, 14607510, 25883640, 25456080, 15309840, 5832288, 1403136, 204288, 16128, 512;
%F Formula
The row polynomials satisfy (16*n^3+52*n-32*n*x-16*n*x^2)*P(n,x) = (40*n^3+120*n^2+198*n+118+(16*n^3+48*n^2-60*n-92)*x-(88*n+88)*x^2-(16*n+16)*x^3)*P(n-1,x) + (52*n^2+52*n-39+(80*n^2+80*n-268)*x+(16*n^2+16*n-384)*x^2-144*x^3-16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((40*n^3 + 120*n^2 + 198*n + 118 + (16*n^3 + 48*n^2 - 60*n - 92)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*P[n - 1, x] + (52*n^2 + 52*n - 39 + (80*n^2 + 80*n - 268)*x + (16*n^2 + 16*n - 384)*x^2 - 144*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 + 52*n - 32*n*x - 16*n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((40*n^3 + 120*n^2 + 198*n + 118 + (16*n^3 + 48*n^2 - 60*n - 92)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*v[i - 1] + (52*n^2 + 52*n - 39 + (80*n^2 + 80*n - 268)*x + (16*n^2 + 16*n - 384)*x^2 - 144*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 + 52*n - 32*n*x - 16*n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023