Real-rooted atlas

A116414 → basis A007318 → basis A013610

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
23336978
318030616227675
498122681863648815841
5534615633174969396243024350544
629133103032146286106920425258748729437373
7158760658692113432410521905686201793343061821873784725
886516141193368343162939535264407152755620719523104976656132750406
94714686253354235901400878245028651244863531786312492144278186435429419683283399479
102569263315379858840506083161832613660615985539901377617874788453852688104516731180980590492452344093
111400117409238631582714718726469040422552965640804106608632223205220085101418822320522038381850389723417714721220898400
127629925415501818404178447193373443160312044032117590393323027042517589544411640683664386091886589636382013835180712754584531441183631053321
13415792002632528298573115429615056246052564056351597822660355770343098262356124176142818745752574179485221684579396535075106004910514844145239815943231589016784689
14226585425331911380963047366284280441719309420144271564827535830714186184562565920088018160905879604876138117179327063951789671167390074134305769521720274517133923132478296913750257902238
Technical details
Node id
n-338253c669fb654f · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-71237460ed9af907

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,        3;
  2:      33,       36,        9;
  3:     180,      306,      162,       27;
  4:     981,     2268,     1863,      648,       81;
  5:    5346,    15633,    17496,     9396,     2430,      243;
  6:   29133,   103032,   146286,   106920,    42525,     8748,      729;
  7:  158760,   658692,  1134324,  1052190,   568620,   179334,    30618,    2187;
  8:  865161,  4119336,  8343162,  9395352,  6440715,  2755620,   719523,  104976,   6561;
  9: 4714686, 25335423, 59014008, 78245028, 65124486, 35317863, 12492144, 2781864, 354294, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -3*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-3*P[n - 2, x] + (6 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -3*v[i - 2] + (6 + 3*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