Real-rooted atlas

A110441 → basis A013609 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16612
2347236142
31926246482161680
41084475275605184129619876
56120337687257676032388807776235152
634552229824622944881280686880279936466562782072
719507215187204971456887328093312005785344195955227993632914560
81101328982310437695168813058561078531209013939246376064134369281679616389410576
96217824625091522751667206962595841116291456117638438481517363235831808090699264100776964607097792
103510428839271564819506648965665946112106596000001357577625611856595968701407641626907448326046617606046617654506352352
11198190080244190131213509949824443071468809572069376014305948876815096388608011250739814458047528960197522841603990767616362797056644862032640
121118931904150551170569180977702433557068800081884784384014055295426561740427057152156679955251210179480729604655895552001423373783042612138803221767823367629331686976
1363172112649216049113661416946483224756684963846735069803520130601453414401857381182668819618344861696153923906949128861721200640363958006579210100270039041697890222081306069401690262256178432
143566540377656075598028840543601817601786788639129653626734111744115960231784448186322888894464226019341025280208028517193728144593492262912747231328419842784539964211270745425920001097098297344783641640961067888410767232
Technical details
Node id
n-40176f9b9853cb34 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-edc8f1d0e47ba8f5

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:       6,        6;
  2:      34,       72,        36;
  3:     192,      624,       648,       216;
  4:    1084,     4752,      7560,      5184,       1296;
  5:    6120,    33768,     72576,     76032,      38880,       7776;
  6:   34552,   229824,    622944,    881280,     686880,     279936,     46656;
  7:  195072,  1518720,   4971456,   8873280,    9331200,    5785344,   1959552,    279936;
  8: 1101328,  9823104,  37695168,  81305856,  107853120,   90139392,  46376064,  13436928,  1679616;
  9: 6217824, 62509152, 275166720, 696259584, 1116291456, 1176384384, 815173632, 358318080, 90699264, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (6 + 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] = 1; v[2] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (6 + 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