Real-rooted atlas

A116412 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 + 5\,x^{2} + 26\,x^{3} + n + 4\,n x - 15\,n x^{2} - 39\,n x^{3} + 4\,n^{2} x + 10\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-x + 6\,x^{2} + 51\,x^{3} + 52\,x^{4} - n x - 12\,n x^{2} - 33\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 11\,x^{3} + 23\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2181322
3112424398
411687190142436
51201485057974691940
612422510522589319215498632
7128318189563981226112391511638408
813242730981336035096549554703016897170896
91365524725248678346917890223654717549455807760400
101406936840425671745284777658633049871206462161843183383392
111448509507683643318211109685255897039924814019617235426760876115054368
12148102312790104418586320232224365403181301046017847484160464548502096201060166984256
1315212121675315314597744544813881494855336300069634303157760100463015342304799176640564298045760
14156141721460217217155408881044613130128090008289191881300298749893329712292244074493108593012219322931326151552
Technical details
Node id
n-2e88d3670212e33f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6249641443f0dcab

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: 1,  4;
  2: 1,  8,  13;
  3: 1, 12,  42,   43;
  4: 1, 16,  87,  190,   142;
  5: 1, 20, 148,  505,   797,   469;
  6: 1, 24, 225, 1052,  2589,  3192,   1549;
  7: 1, 28, 318, 1895,  6398, 12261,  12391,   5116;
  8: 1, 32, 427, 3098, 13360, 35096,  54955,  47030,  16897;
  9: 1, 36, 552, 4725, 24867, 83469, 178902, 236547, 175494, 55807;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n+1+(4*n^2+4*n)*x+(10*n^2-15*n+5)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - ((n+1)*x+(12*n-6)*x^2+(33*n-51)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (x^2+11*x^3+23*x^4+13*x^5)*d^2/dx^2 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[Together[((n + 1 + (4*n^2 + 4*n)*x + (10*n^2 - 15*n + 5)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - ((n + 1)*x + (12*n - 6)*x^2 + (33*n - 51)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (x^2 + 11*x^3 + 23*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(n + 1)]];
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] = ((n + 1 + (4*n^2 + 4*n)*x + (10*n^2 - 15*n + 5)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - ((n + 1)*x + (12*n - 6)*x^2 + (33*n - 51)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (x^2 + 11*x^3 + 23*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(n + 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

Candidate equalities ?

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