Real-rooted atlas

A103327 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,n\right)\,P_{n}(x) = \left(-3 + 6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(12\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
29361661
32721624064547
4811080216013442564921
5243486015120161286912102444287
67292041290720145152103680337924096398581
721878164848988810886401140480608256159744163843587227
865613149282449440718502410264320790732833546247372806553632285041
91968311809801154736043110144800616968302694450319360176947203342336262144290565367
1059049433026051963120240185088560431872747242496603832320300810240902430721494220810485762615088301
111771471558893622517352012609717123602776320597793996861590896644060938240171461836844826624066060288419430423535794707
1253144155269864945728784630485856021616657920435535626245543180697646294695936257192755209413591040217998950428940697616777216211822152361
13159432319344452438688904803026332108812249439488029398654771245137328537646294695936032406287155215532425216050139758592104186511361258291200671088641906399371247
144782969669615660154755619201403117614086614697323521861914802176338529964032041665226342403564691587072214347467980890251565465626046627840049073356800543581798426843545617157594341221
Technical details
Node id
n-50ed9595bffee798 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-05f5318533cf2c44

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:     3,       4;
  2:     9,      36,       16;
  3:    27,     216,      240,       64;
  4:    81,    1080,     2160,     1344,      256;
  5:   243,    4860,    15120,    16128,     6912,     1024;
  6:   729,   20412,    90720,   145152,   103680,    33792,     4096;
  7:  2187,   81648,   489888,  1088640,  1140480,   608256,   159744,    16384;
  8:  6561,  314928,  2449440,  7185024, 10264320,  7907328,  3354624,   737280,   65536;
  9: 19683, 1180980, 11547360, 43110144, 80061696, 83026944, 50319360, 17694720, 3342336, 262144;
%F Formula
The row polynomials satisfy (2*n-1)*P(n,x) = (6*n-3+4*n*x)*P(n-1,x) + (12*x+4*x^2)*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[((6*n - 3 + 4*n*x)*P[n - 1, x] + (12*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];
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] = ((6*n - 3 + 4*n*x)*v[i - 1] + (12*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023