Real-rooted atlas

A257626 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-54 + 36\,x - 54\,n + 9\,n x^{2} - 36\,n^{2} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 18\,x^{2} - 18\,n x + 9\,n x^{2} + 9\,n x^{3} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(6 + 15\,n + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
166
210872180
3140428086484860
41587667392476285184136080
516621212849841927476664848388804082400
616592042140344057824280428068808296020279936132269760
716052580326527848144525708019270094408163196209631548019595524629441600

All 15 rows

Technical details
Node id
n-5f3bc25b4b68067b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-96547810142a253d

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:          6;
  2:        108,          72;
  3:       1404,        2808,          648;
  4:      15876,       67392,        47628,          5184;
  5:     166212,     1284984,      1927476,        664848,         38880;
  6:    1659204,    21403440,     57824280,      42806880,       8296020,        279936;
  7:   16052580,   326527848,   1445257080,    1927009440,     816319620,      96315480,      1959552;
  8:  152031492,  4688858016,  31964483412,   69372339840,   53274139020,   14066574048,   1064220444,    13436928;
  9: 1418671908, 64475996760, 648266139756, 2143707652656, 2679634565820, 1296532279512, 225665988660, 11349375264, 90699264;
%F Formula
The row polynomials satisfy n*P(n,x) = -(54*n+54+(36*n^2-36)*x+(9*n^3-9*n)*x^2)*P(n-2,x) - ((18*n+18)*x+(9*n^2-9*n-18)*x^2-(9*n^2+9*n)*x^3)*d/dx P(n-2,x) + (15*n+6+6*n^2*x)*P(n-1,x) + (3*n*x-3*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n + 54 + (36*n^2 - 36)*x + (9*n^3 - 9*n)*x^2)*P[n - 2, x] - ((18*n + 18)*x + (9*n^2 - 9*n - 18)*x^2 - (9*n^2 + 9*n)*x^3)*D[P[n - 2, x], x] + (15*n + 6 + 6*n^2*x)*P[n - 1, x] + (3*n*x - 3*n*x^2)*D[P[n - 1, x], x])/(n)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n + 54 + (36*n^2 - 36)*x + (9*n^3 - 9*n)*x^2)*v[i - 2] - ((18*n + 18)*x + (9*n^2 - 9*n - 18)*x^2 - (9*n^2 + 9*n)*x^3)*deriv(v[i - 2], x) + (15*n + 6 + 6*n^2*x)*v[i - 1] + (3*n*x - 3*n*x^2)*deriv(v[i - 1], x))/(n)); 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

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