Real-rooted atlas

A308737 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-12 + 6\,n - 14\,n x + 2\,n x^{2} + 9\,n^{2} x - 3\,n^{2} x^{2} - n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4 - 4\,x + n - 3\,n x - 3\,n x^{2} + n x^{3} + n^{2} x - n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)
\(P_{2}(x) = 8 + 14\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
281422
3176245124
434192282124632
567516114610123153056
6132129038405100318076214304
726130901153520460191559186178565472
8518719632298716808953064428250464088294784
9103116424861002294323583303438962005646548092071310464
10205636918221328688296129024015480361203888589608165816204705766656
114105819905531851966440430101061929006020490393156016585654095104504525164800
128202180312135201054073801351746022708224264920042162556012164130450516099112298292109049856
131639539332432444941441902040551885778575601059707881038066487298824536043260118941902359164212979469757952
14327808520987668336374883081171484602530370703936092164498356723795465962342869101030858403067006855376364587382013257728
Technical details
Node id
n-f9ec10621819e3f8 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-ffeaa4374b687cd3

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    0;
  1:    3;
  2:    8,    14;
  3:   17,    62,    45;
  4:   34,   192,   282,    124;
  5:   67,   516,  1146,   1012,    315;
  6:  132,  1290,  3840,   5100,   3180,    762;
  7:  261,  3090, 11535,  20460,  19155,   9186,   1785;
  8:  518,  7196, 32298,  71680,  89530,  64428,  25046,  4088;
  9: 1031, 16424, 86100, 229432, 358330, 343896, 200564, 65480, 9207;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (6*n-12-(n^3-9*n^2+14*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-1,x) + (n-4+(n^2-3*n-4)*x-3*n*x^2-(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 3, P(2,x) = 8 + 14*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[2, x_] := 8 + 14*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n - 12 - (n^3 - 9*n^2 + 14*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] + (n - 4 + (n^2 - 3*n - 4)*x - 3*n*x^2 - (n^2 - n)*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; v[3] = 8 + 14*x; for(n = 3, N, my(i = n + 1); v[i] = ((6*n - 12 - (n^3 - 9*n^2 + 14*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] + (n - 4 + (n^2 - 3*n - 4)*x - 3*n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 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

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.