Real-rooted atlas

A207823 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10\,n - 2\,n x - 6\,n x^{2} + 6\,n^{3}\right)\,P_{n}(x) = \left(19 - 11\,x - 27\,x^{2} - 9\,x^{3} + 49\,n + 7\,n x - 27\,n x^{2} - 9\,n x^{3} + 45\,n^{2} + 27\,n^{2} x + 15\,n^{3} + 9\,n^{3} x\right)\,P_{n-1}(x) + \left(-14 - 69\,x - 97\,x^{2} - 51\,x^{3} - 9\,x^{4} + 21\,n + 30\,n x + 9\,n x^{2} + 21\,n^{2} + 30\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18311
254489111
3334489216271066
4196140562943864819905
511124299043063614742324024389889
6615792042882720791846806642011664729801439
73347181323063217566019192959728102792074082421877047764
81793826823752016148835175899601158219046811521117557139968656161297569
99505540497563651133821801472588101197050406282303321106008431276447239219683528341815
10499076992934037207624943101151411760111602326572291139231523272290594288161794261574640590494519792271
11260030952169671043249545435608534696220961885206074285765823997118664149040988237417383059344245519631217714738419829886
12134606322196545179923130328208660605934480779282892456997864622444886692214206472045606723023895149826996021369833117006112531441324803159761
136929367956541931760511932176444044154773633386005267132406152692326934588583191202515734093601010292960602919244779058484491567578348665526986415943232732930118221
14355003744143006879721441169359979157276565295613644404406042945114403037440435177132106527737971192801327483423035472968318240122821979049223474483442652953472178564176478296922900070833215
Technical details
Node id
n-49b042466eca8b16 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d8dc46745c680172

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:       8,        3;
  2:      54,       48,         9;
  3:     334,      489,       216,        27;
  4:    1961,     4056,      2943,       864,        81;
  5:   11124,    29904,     30636,     14742,      3240,      243;
  6:   61579,   204288,    272079,    184680,     66420,    11664,      729;
  7:  334718,  1323063,   2175660,   1919295,    972810,   279207,    40824,    2187;
  8: 1793826,  8237520,  16148835,  17589960,  11582190,  4681152,  1117557,  139968,   6561;
  9: 9505540, 49756365, 113382180, 147258810, 119705040, 62823033, 21106008, 4312764, 472392, 19683;
%F Formula
The row polynomials satisfy (6*n^3+10*n-2*n*x-6*n*x^2)*P(n,x) = (15*n^3+45*n^2+49*n+19+(9*n^3+27*n^2+7*n-11)*x-(27*n+27)*x^2-(9*n+9)*x^3)*P(n-1,x) + (21*n^2+21*n-14+(30*n^2+30*n-69)*x+(9*n^2+9*n-97)*x^2-51*x^3-9*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[((15*n^3 + 45*n^2 + 49*n + 19 + (9*n^3 + 27*n^2 + 7*n - 11)*x - (27*n + 27)*x^2 - (9*n + 9)*x^3)*P[n - 1, x] + (21*n^2 + 21*n - 14 + (30*n^2 + 30*n - 69)*x + (9*n^2 + 9*n - 97)*x^2 - 51*x^3 - 9*x^4)*D[P[n - 1, x], x])/(6*n^3 + 10*n - 2*n*x - 6*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] = ((15*n^3 + 45*n^2 + 49*n + 19 + (9*n^3 + 27*n^2 + 7*n - 11)*x - (27*n + 27)*x^2 - (9*n + 9)*x^3)*v[i - 1] + (21*n^2 + 21*n - 14 + (30*n^2 + 30*n - 69)*x + (9*n^2 + 9*n - 97)*x^2 - 51*x^3 - 9*x^4)*deriv(v[i - 1], x))/(6*n^3 + 10*n - 2*n*x - 6*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