Real-rooted atlas

A290315 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 3\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
31216129
4120284661471
5168059522872224110729
630240146064116336210807061314427
76652804130304480465615310721322282160111265701
8172972801325577602087916481014226401627108076050065301477107439
95189184004766423040965733350465564590081661273440151922560415512019648123316504721
1017643225600189945688320477104352768427142449920155184721088231552792001304454880219937765901011291502224563
116704425728008312940195840251758769287682851368778752013932826787712305003812556828927371536010582773376114099580177104179955783163629
12281585880576003964745430374401416739710159360196772904831513612358092705000963702291120254725264248459500833442575745608252278255258431201253139415471210121890631
131295295050649600204713746771968008482990975239168014104633929530572811007156859236326442972260451182592849795237798169682654239106252836535095210096625573062848296725994415942721410048406225261049
14647647525324800001137836608834867200539097638321424384010526301141109180416994460397370678425648792748976529569281273432382738384384174943687416639232120878441089156803826151402957924646064834896149861612884782914133404608947510526827
15349729663675392000067743171630799257600362686450980154798080818728332127133515776917174602122767686656550013817926747713536182275399113694249728334468681092401827843313822529783058496167401647824693760388026635714235233991097650496754186636681434884812939051077157786881781
Technical details
Node id
n-4ebd12eca6adc5e0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4329a0349930abbf

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:           2,            1;
   3:          12,           16,            1;
   4:         120,          284,           66,            1;
   5:        1680,         5952,         2872,          224,            1;
   6:       30240,       146064,       116336,        21080,          706,           1;
   7:      665280,      4130304,      4804656,      1531072,       132228,        2160,          1;
   8:    17297280,    132557760,    208791648,    101422640,     16271080,      760500,       6530,        1;
   9:   518918400,   4766423040,   9657333504,   6556459008,   1661273440,   151922560,    4155120,    19648,     1;
  10: 17643225600, 189945688320, 477104352768, 427142449920, 155184721088, 23155279200, 1304454880, 21993776, 59010, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-6+(2*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 6 + (2*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (4*n - 6 + (2*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x)); 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. [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