Real-rooted atlas

A144699 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1516
22516142
3125171391336
462515267868613024
5312512281116063046181130240
6156259243614264770792107673721332640
778125663991155314512795713800713602175513991680
839062546089461551995219555438981849418804941163681522151891840
919531253116930114551123226508316820569907467897410880202436736830571726485760
10976562520654705612991023333284755200370687384219131685444363093143961651211418456128110897286400
11488281251346704211111591868913796996259759639906178450798023341627841371826569447141733100333511327122711174356582400
1224414062586661627669292805455841535156968687848881897592791727276449462116723612955381887615537753999742398710107180462455812964061900800
13122070312555170804721754490335098434502115762612059915067635171855720984151286417102170850169939971649793137152438808235193531303655303254706649133153353114214400
14610351562534811467207659986528070594381405328218415633446834260129245487166050029617603012452316311729613963247850036278307710903521854848736724262513042215176984764599828411013709170073600
Technical details
Node id
n-75da9dd910387754 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-edf8fec256ab12fb

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:       5,        1;
  2:      25,       16,         1;
  3:     125,      171,        39,         1;
  4:     625,     1526,       786,        86,         1;
  5:    3125,    12281,     11606,      3046,       181,        1;
  6:   15625,    92436,    142647,     70792,     10767,      372,       1;
  7:   78125,   663991,   1553145,   1279571,    380071,    36021,     755,      1;
  8:  390625,  4608946,  15519952,  19555438,   9818494,  1880494,  116368,   1522,    1;
  9: 1953125, 31169301, 145511232, 265083168, 205699074, 67897410, 8802024, 367368, 3057, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + n*x)*v[i - 1] + (x - 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 - 1, v[i])))};
T(9)

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