Real-rooted atlas

A257622 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n\right)\,P_{n}(x) = \left(-28 + 40\,x - 10\,x^{2} - 84\,n + 96\,n x - 9\,n x^{2} - 72\,n^{2} x + 54\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 18\,x^{2} - 6\,x^{3} - 36\,n x + 45\,n x^{2} - 9\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 8\,x + 33\,n - 24\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2563288
355211041921848
446962208014088102441888
53696834289651434414787251201047200
6278232461193613372944922387213911602457628902720
720377365669217628453473637937964814173044012226416114688876715840

All 15 rows

Technical details
Node id
n-d5ced0bffca38ecf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-afbfd84852f44762

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:         4;
  2:        56,         32;
  3:       552,       1104,         192;
  4:      4696,      22080,       14088,         1024;
  5:     36968,     342896,      514344,       147872,         5120;
  6:    278232,    4611936,    13372944,      9223872,      1391160,        24576;
  7:   2037736,   56692176,   284534736,    379379648,    141730440,     12226416,      114688;
  8:  14673752,  656582144,  5314678584,  12140148736,   8857797640,   1969746432,   102716264,    524288;
  9: 104551272, 7299509040, 90758032344, 328880903904, 411101129880, 181516064688, 25548281640, 836410176, 2359296;
%F Formula
The row polynomials satisfy (3*n-2)*P(n,x) = -(84*n+28+(72*n^2-96*n-40)*x+(27*n^3-54*n^2+9*n+10)*x^2)*P(n-2,x) - ((36*n+12)*x+(27*n^2-45*n-18)*x^2-(27*n^2-9*n-6)*x^3)*d/dx P(n-2,x) + (33*n-10+(18*n^2-24*n+8)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*P[n - 2, x] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*D[P[n - 2, x], x] + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x])/(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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n + 28 + (72*n^2 - 96*n - 40)*x + (27*n^3 - 54*n^2 + 9*n + 10)*x^2)*v[i - 2] - ((36*n + 12)*x + (27*n^2 - 45*n - 18)*x^2 - (27*n^2 - 9*n - 6)*x^3)*deriv(v[i - 2], x) + (33*n - 10 + (18*n^2 - 24*n + 8)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x))/(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.