Real-rooted atlas

A100862 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(2 + 6\,x^{2} + 8\,x^{3} + 3\,n x - 10\,n x^{2} - 12\,n x^{3} + n^{2} x + 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x + 2\,x^{2} + 22\,x^{3} + 16\,x^{4} - 4\,n x^{2} - 12\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x) + \left(4\,x^{3} + 8\,x^{4} + 4\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
215511
319211445
4114578643201
5120125320355142963
6127240925171014974994899
71354202275623089676475185026253
81446864984189353987246858287727193147345
9154106299965046314578224817824602413133729186862083
101651575186901214854619791069005152295013040556155451231095238459
1117722553300026944513095393955413760056092919756997265295942953807832957037
1219031355555055836033903541299259532352210530046005667827038069559145821302430355214117209
131104425189804109252081364143872111112151682425638069636539789034689697921020157673564699113176461433320515
141119564214023120350331830228410632722141193738310888968091987797812250357006921357173111178439808379609349542299079867008979
Technical details
Node id
n-da15dfce718ba206 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-83abbbf5c563f720

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,  2;
  2: 1,  5,    5;
  3: 1,  9,   21,   14;
  4: 1, 14,   57,   86,    43;
  5: 1, 20,  125,  320,   355,    142;
  6: 1, 27,  240,  925,  1710,   1497,    499;
  7: 1, 35,  420, 2275,  6230,   8967,   6475,   1850;
  8: 1, 44,  686, 4984, 18935,  39872,  46858,  28772,   7193;
  9: 1, 54, 1062, 9996, 50463, 145782, 248178, 246024, 131337, 29186;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2+(n^2+3*n)*x+(4*n^2-10*n+6)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (2*x+(4*n-2)*x^2+(12*n-22)*x^3+(8*n-16)*x^4)*d/dx P(n-1,x) + (4*x^3+8*x^4+4*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[((2 + (n^2 + 3*n)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (2*x + (4*n - 2)*x^2 + (12*n - 22)*x^3 + (8*n - 16)*x^4)*D[P[n - 1, x], x] + (4*x^3 + 8*x^4 + 4*x^5)*D[P[n - 1, x], {x, 2}])/(2)]];
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] = ((2 + (n^2 + 3*n)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (2*x + (4*n - 2)*x^2 + (12*n - 22)*x^3 + (8*n - 16)*x^4)*deriv(v[i - 1], x) + (4*x^3 + 8*x^4 + 4*x^5)*deriv(deriv(v[i - 1], x), x))/(2)); 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