Real-rooted atlas

A207606 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(5 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 43 + 114\,x + 72\,x^{2}\)
\(P_{1}(x) = 206 + 816\,x + 1044\,x^{2} + 432\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04311472229
120681610444322498
29875202100448424259227249
3472931116803881019526350415552297241
422658178752578592983664926640458784933123242402
510856199859238850848287920104716807838208321926455987235369181
65201475465574248383806376644010115928010156233663032256221149443359232385818589
72492174294501601531002604595745608779233601106929152921316032488208384149485824201553924208635298

All 15 rows

Technical details
Node id
n-3f3ee0e69ccb4b94 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-9039c923d279ed7f

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:       43,       114,         72;
  1:      206,       816,       1044,         432;
  2:      987,      5202,      10044,        8424,        2592;
  3:     4729,     31116,      80388,      101952,       63504,       15552;
  4:    22658,    178752,     578592,      983664,      926640,      458784,        93312;
  5:   108561,    998592,    3885084,     8287920,    10471680,     7838208,      3219264,       559872;
  6:   520147,   5465574,   24838380,    63766440,   101159280,   101562336,     63032256,     22114944,     3359232;
  7:  2492174,  29450160,  153100260,   459574560,   877923360,  1106929152,    921316032,    488208384,   149485824,    20155392;
  8: 11940723, 156738270,  917363880,  3152707920,  7045904880, 10700623584,  11185122816,   7946823168,  3673320192,   997691904,  120932352;
  9: 57211441, 825885528, 5374148760, 20808148320, 53267848560, 94671618048, 119208039552, 106356644352, 65898054144, 27008225280, 6590813184, 725594112;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 43 + 114*x + 72*x^2, P(1,x) = 206 + 816*x + 1044*x^2 + 432*x^3.
%t Mathematica
P[0, x_] := 43 + 114*x + 72*x^2;
P[1, x_] := 206 + 816*x + 1044*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 43 + 114*x + 72*x^2; v[2] = 206 + 816*x + 1044*x^2 + 432*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (5 + 6*x)*v[i - 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

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