Real-rooted atlas

A049403 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-x - 3\,x^{2} - 2\,x^{3} - n - 4\,n x - 5\,n x^{2} - 2\,n x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 3\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
00347
103121025
20015402681
300159015076331
40001054205462321303
5000105840210021287645937
60000945504098288352262026785
7000094594503150047880343809496133651
800000103956930018018022968014410035696669351
90000010395124740519750105336011345406267361401523609673
100000001351351081080351351059716805619900278428856850419674097
1100000013513518918909459450239639403441438028516488127538322390480113525595
120000000202702518918900737837101567566001966965001461751205969292010349536664400311
13000000020270253243240018918900057513456010324314001140659520765229920286857600462067364070168161
14000000003445942536756720016724307604263779520668768100066266054404059118560140753689621179931225330978113
Technical details
Node id
n-601d114a828de4d8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a5e5e7677b9b26e0

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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: 0, 3,  4;
  1: 0, 3, 12,  10;
  2: 0, 0, 15,  40,  26;
  3: 0, 0, 15,  90, 150,    76;
  4: 0, 0,  0, 105, 420,   546,    232;
  5: 0, 0,  0, 105, 840,  2100,   2128,    764;
  6: 0, 0,  0,   0, 945,  5040,   9828,   8352,    2620;
  7: 0, 0,  0,   0, 945,  9450,  31500,  47880,   34380,    9496;
  8: 0, 0,  0,   0,   0, 10395,  69300, 180180,  229680,  144100,  35696;
  9: 0, 0,  0,   0,   0, 10395, 124740, 519750, 1053360, 1134540, 626736, 140152;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n+(4*n+1)*x+(5*n+3)*x^2+(2*n+2)*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3*x + 4*x^2.
%t Mathematica
P[0, x_] := 3*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n + 1)*x + (5*n + 3)*x^2 + (2*n + 2)*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n + (4*n + 1)*x + (5*n + 3)*x^2 + (2*n + 2)*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(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