Real-rooted atlas

A123519 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 6\,x - 3\,x^{2} + 13\,n - 4\,n x - 2\,n x^{2} + 9\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(30 - 6\,x - 19\,x^{2} - 4\,x^{3} + 53\,n + 16\,n x - 10\,n x^{2} - 2\,n x^{3} + 27\,n^{2} + 14\,n^{2} x + 4\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-24\,x - 38\,x^{2} - 16\,x^{3} - 2\,x^{4} + 24\,n + 32\,n x + 8\,n x^{2} + 6\,n^{2} + 8\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 25 + 22\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02522451
1117164688357
25251020684184162429
322895710535623364643216187
497732983436152221127120112064106175
54106514845622111217512079120201602624128687785
6170393712344126055212295527199202573445420860162564410585
7699869332313068174167914016571198426465607817601402881356851228049103
82850625151581103538940447694944410046562342793689689602255872352512302081024177134251
91152943767885004177777308272997208272705840185674496876592642856806462538248652806656020481111984333
10463528052994139728693882441499067240170750704013514252807604513283056040968665728016788480208384014540840966944889109
11185397705130366618241574227407955367360101816606409192802752601451136028809492481007209728252679680438958084939776315392819243180826563
1273821602156138635701950739256041028943680583096299205917981932844181348480246087690241025412326431710581767131238401122713601155481667993616384267430810359
1329278624812394425608090055720240206528172320322790973280363934340992305499026560194193136896940944744963467764326496111431681958195200281841664267223041458176327681650524999889
1411571840305101276004272409919697360101812710064017359417153602153249659008200847652544014357625413127939832440323402524364801123698667522820999168052529684486962708486119424031129606553610155154234673
Technical details
Node id
n-a5146145a4649388 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-51272ecb239e904d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:       25,       22,         4;
  1:      117,      164,        68,         8;
  2:      525,     1020,       684,       184,        16;
  3:     2289,     5710,      5356,      2336,       464,        32;
  4:     9773,    29834,     36152,     22112,      7120,      1120,       64;
  5:    41065,   148456,    221112,    175120,     79120,     20160,     2624,      128;
  6:   170393,   712344,   1260552,   1229552,    719920,    257344,    54208,     6016,     256;
  7:   699869,  3323130,   6817416,   7914016,   5711984,   2646560,   781760,   140288,   13568,    512;
  8:  2850625, 15158110,  35389404,  47694944,  41004656,  23427936,  8968960,  2255872,  352512,  30208,  1024;
  9: 11529437, 67885004, 177777308, 272997208, 272705840, 185674496, 87659264, 28568064, 6253824, 865280, 66560, 2048;
%F Formula
The row polynomials satisfy (2*n^3+9*n^2+13*n+6-(4*n+6)*x-(2*n+3)*x^2)*P(n,x) = (4*n^3+27*n^2+53*n+30+(2*n^3+14*n^2+16*n-6)*x-(10*n+19)*x^2-(2*n+4)*x^3)*P(n-1,x) + (6*n^2+24*n+(8*n^2+32*n-24)*x+(2*n^2+8*n-38)*x^2-16*x^3-2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 25 + 22*x + 4*x^2.
%t Mathematica
P[0, x_] := 25 + 22*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 + 27*n^2 + 53*n + 30 + (2*n^3 + 14*n^2 + 16*n - 6)*x - (10*n + 19)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + (6*n^2 + 24*n + (8*n^2 + 32*n - 24)*x + (2*n^2 + 8*n - 38)*x^2 - 16*x^3 - 2*x^4)*D[P[n - 1, x], x])/(2*n^3 + 9*n^2 + 13*n + 6 - (4*n + 6)*x - (2*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 25 + 22*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 27*n^2 + 53*n + 30 + (2*n^3 + 14*n^2 + 16*n - 6)*x - (10*n + 19)*x^2 - (2*n + 4)*x^3)*v[i - 1] + (6*n^2 + 24*n + (8*n^2 + 32*n - 24)*x + (2*n^2 + 8*n - 38)*x^2 - 16*x^3 - 2*x^4)*deriv(v[i - 1], x))/(2*n^3 + 9*n^2 + 13*n + 6 - (4*n + 6)*x - (2*n + 3)*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