Real-rooted atlas

A078812 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(5 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
25611
3246036120
41154385402161309
5551282052924320129614279
62640169684284053136324007776155760
71264997860310716518400479520233280466561699081
8606055482262097900440316054756004035744163296027993618534131
929037630069001346814034084800533174405279904032332608111974401679616202176360
101391275162285308328420024682968046562040057986409647682432024970291275582720100776962205405829
116665999864834005003240401699768800375576264056402438405830973568409826304018744514565038848006046617624057287759
12319387204561844642937236400112539585602851180560050155930944625195065605522745369633886252800137560550403325639680362797056262424759520
1315302760123860712401692296474472193442400206327016720416210244480607702144896647156044800498921534720271593907200991040624642176782336021767823362862615066961
14733199285123923373429599401476045125104190414362839324003268857391776547325268480068267656396806343657689600433774268928021217581158407031006945281414908518401306069401631226340977051
153512968824639748111805374011331082760025855680968259889670424545780308800463717056297606632618826240072179960751360594790656768003653614265241616224284344320492388164403291424858112078364164096340627135680600
Technical details
Node id
n-3b0e200027203be8 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-83732280cb4a5f8a

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,        6;
   3:      24,       60,       36;
   4:     115,      438,      540,       216;
   5:     551,     2820,     5292,      4320,      1296;
   6:    2640,    16968,    42840,     53136,     32400,      7776;
   7:   12649,    97860,   310716,    518400,    479520,    233280,     46656;
   8:   60605,   548226,  2097900,   4403160,   5475600,   4035744,   1632960,    279936;
   9:  290376,  3006900, 13468140,  34084800,  53317440,  52799040,  32332608,  11197440,  1679616;
  10: 1391275, 16228530, 83284200, 246829680, 465620400, 579864096, 476824320, 249702912, 75582720, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+6*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 6*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 6*x;
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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 6*x; for(n = 3, N, my(i = n); 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, v[i])))};
T(10)

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