Real-rooted atlas

A100862 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + 2\,x^{3} + 2\,n + 3\,n x + 2\,n x^{2} + n^{2} x\right)\,P_{n}(x) = \left(6 + 8\,x + 6\,x^{2} + 8\,x^{3} + 2\,x^{4} + 7\,n + 15\,n x + 7\,n x^{2} - n^{2} x^{2} - n^{3} - n^{3} x\right)\,P_{n-1}(x) + \left(4 + 12\,x + 16\,x^{2} + 12\,x^{3} + 4\,x^{4} + 6\,n + 16\,n x + 14\,n x^{2} + 4\,n x^{3} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2107118
3353912187
412920099181447
54979956952052512418
6199649174485188537533113692
78325244092765715330437563042180769
835965122488166474115612438559100994521494541
91605236233859905588313063980971104391743014946313133296
107386543223655587290557989703378900119088925231531290216075120489814
11349750716953959348731903969421527377625118126473195654533445532953025881137994651
12170124859072124820810436926869667021476565011030778036725073785723410583108690041251021955339947
1384882345493967851125089251318084885391647301370985481640392564601103812423179690941990560136565549911716787493118
144338392562736488965758559174112149997587124425591298521178666397245849012625551512710477773865261435775742079357189133149418162208
Technical details
Node id
n-22a18b3ce90d3ed6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ca83066fdcf05149

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:      3,      1;
  2:     10,      7,      1;
  3:     35,     39,     12,      1;
  4:    129,    200,     99,     18,      1;
  5:    497,    995,    695,    205,     25,      1;
  6:   1996,   4917,   4485,   1885,    375,     33,     1;
  7:   8325,  24409,  27657,  15330,   4375,    630,    42,    1;
  8:  35965, 122488, 166474, 115612,  43855,   9100,   994,   52,  1;
  9: 160523, 623385, 990558, 831306, 398097, 110439, 17430, 1494, 63, 1;
%F Formula
The row polynomials satisfy (2*n+2+(n^2+3*n+2)*x+2*n*x^2+2*x^3)*P(n,x) = -(n^3-7*n-6+(n^3-15*n-8)*x+(n^2-7*n-6)*x^2-8*x^3-2*x^4)*P(n-1,x) + (2*n^2+6*n+4+(4*n^2+16*n+12)*x+(2*n^2+14*n+16)*x^2+(4*n+12)*x^3+4*x^4)*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[Together[(-(n^3 - 7*n - 6 + (n^3 - 15*n - 8)*x + (n^2 - 7*n - 6)*x^2 - 8*x^3 - 2*x^4)*P[n - 1, x] + (2*n^2 + 6*n + 4 + (4*n^2 + 16*n + 12)*x + (2*n^2 + 14*n + 16)*x^2 + (4*n + 12)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n + 2 + (n^2 + 3*n + 2)*x + 2*n*x^2 + 2*x^3)]];
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^3 - 7*n - 6 + (n^3 - 15*n - 8)*x + (n^2 - 7*n - 6)*x^2 - 8*x^3 - 2*x^4)*v[i - 1] + (2*n^2 + 6*n + 4 + (4*n^2 + 16*n + 12)*x + (2*n^2 + 14*n + 16)*x^2 + (4*n + 12)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n + 2 + (n^2 + 3*n + 2)*x + 2*n*x^2 + 2*x^3)); 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

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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.