Real-rooted atlas

A094816 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2325
3915529
427816916193
581378585348651457
6243162039154230200732612341
77296561226803820532706132871957116125
8218725515119070286335380268276129100125137001203329
9656196228581742188470835590593983364255659484921610960113627073
1019683354294269001011267424285716974525000244461278259045568019963986410167525317
1159049127939511908215625263302044210774330763015991023255309374502860006598353587998641012222710781
1217714745467735088055532693463013366790913639607587670850561783295402306784638597342356998595182322110850511231665408545
13531441159432302111001751628177760812919709827607861314651917836351074353544001220478132069263069976644215872219117803225401302061345482196718129
141594323552698648543799817785374454465715132741927091277545660452602091191315743046178973633381418866070014221348061750217613280338866157408820943169267974867817359305653
15478296918972443733852791703596456108725375085435712555337714924483093134663117057239351912239145195987731109206404276307566687003032085656817289590966252069722258768513583236975164805134443910166077
Technical details
Node id
n-ad1152346a0e4273 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-45213710b95c651a

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:     3,      2;
   3:     9,     15,       5;
   4:    27,     81,      69,       16;
   5:    81,    378,     585,      348,       65;
   6:   243,   1620,    3915,     4230,     2007,      326;
   7:   729,   6561,   22680,    38205,    32706,    13287,     1957;
   8:  2187,  25515,  119070,   286335,   380268,   276129,   100125,    13700;
   9:  6561,  96228,  581742,  1884708,  3559059,  3983364,  2556594,   849216,  109601;
  10: 19683, 354294, 2690010, 11267424, 28571697, 45250002, 44461278, 25904556, 8019963, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(n-2)*x^2*P(n-2,x) + (3+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n - 2)*x^2*P[n - 2, x] + (3 + n*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] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n - 2)*x^2*v[i - 2] + (3 + n*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

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