Real-rooted atlas

A081579 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-30 + 27\,x^{2} + 30\,n - 45\,n x - 27\,n x^{2} + 45\,n^{2} x\right)\,P_{n}(x) = \left(-420 - 1530\,x - 972\,x^{2} - 162\,x^{3} + 630\,n + 2025\,n x + 648\,n x^{2} - 150\,n^{2} - 405\,n^{2} x\right)\,P_{n-1}(x) + \left(-100 - 660\,x - 1305\,x^{2} - 702\,x^{3} - 81\,x^{4} + 100\,n + 600\,n x + 945\,n x^{2} + 135\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2241842
310825281441
4480180016203244224
5192010080162008424121537839
6748850400116640108864388804374326646
728224232128706320977184617220166212153092742597

All 15 rows

Technical details
Node id
n-9814cb41048979ea · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11ca45fc8a47c4f6

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      3;
  2:     24,      18;
  3:    108,     252,       81;
  4:    480,    1800,     1620,      324;
  5:   1920,   10080,    16200,     8424,     1215;
  6:   7488,   50400,   116640,   108864,    38880,     4374;
  7:  28224,  232128,   706320,   977184,   617220,   166212,    15309;
  8: 104448, 1014912,  3830976,  7081344,  6687360,  3131784,   673596,   52488;
  9: 380160, 4262400, 19232640, 44696448, 57056400, 39890880, 14696640, 2624400, 177147;
%F Formula
The row polynomials satisfy (30*n-30+(45*n^2-45*n)*x-(27*n-27)*x^2)*P(n,x) = -(150*n^2-630*n+420+(405*n^2-2025*n+1530)*x-(648*n-972)*x^2+162*x^3)*P(n-1,x) + (100*n-100+(600*n-660)*x+(945*n-1305)*x^2+(135*n-702)*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(150*n^2 - 630*n + 420 + (405*n^2 - 2025*n + 1530)*x - (648*n - 972)*x^2 + 162*x^3)*P[n - 1, x] + (100*n - 100 + (600*n - 660)*x + (945*n - 1305)*x^2 + (135*n - 702)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(30*n - 30 + (45*n^2 - 45*n)*x - (27*n - 27)*x^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; for(n = 2, N, my(i = n + 1); v[i] = (-(150*n^2 - 630*n + 420 + (405*n^2 - 2025*n + 1530)*x - (648*n - 972)*x^2 + 162*x^3)*v[i - 1] + (100*n - 100 + (600*n - 660)*x + (945*n - 1305)*x^2 + (135*n - 702)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(30*n - 30 + (45*n^2 - 45*n)*x - (27*n - 27)*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

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.