Real-rooted atlas

A257614 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 5\,n\right)\,P_{n}(x) = \left(42 - 156\,x + 312\,x^{2} - 70\,n + 320\,n x - 835\,n x^{2} - 100\,n^{2} x + 600\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 150\,x^{2} + 120\,x^{3} - 50\,n x + 325\,n x^{2} - 275\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-62 + 128\,x + 45\,n - 160\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-40\,x + 40\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
228836
324448824756
418441171255326419152
51326020324030486053040160574560
693684302232010365240604464046842038419994688
7657836413267762759456403679275201033169403947016896793122624

All 15 rows

Technical details
Node id
n-ba4ac7bddd5cc4ce · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-4f56baeab55bc9ab

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:        2;
  2:       28,          8;
  3:      244,        488,           24;
  4:     1844,      11712,         5532,           64;
  5:    13260,     203240,       304860,        53040,          160;
  6:    93684,    3022320,     10365240,      6044640,       468420,          384;
  7:   657836,   41326776,    275945640,    367927520,    103316940,      3947016,         896;
  8:  4609588,  538022816,   6364810308,  16188810880,  10608017180,   1614068448,    32267116,      2048;
  9: 32277868, 6797383304, 134026870404, 585287010448, 731608763060, 268053740808, 23790841564, 258222944, 4608;
%F Formula
The row polynomials satisfy (5*n-8)*P(n,x) = -(70*n-42+(100*n^2-320*n+156)*x+(125*n^3-600*n^2+835*n-312)*x^2)*P(n-2,x) - ((50*n-30)*x+(125*n^2-325*n+150)*x^2-(125*n^2-275*n+120)*x^3)*d/dx P(n-2,x) + (45*n-62+(50*n^2-160*n+128)*x)*P(n-1,x) + ((25*n-40)*x-(25*n-40)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n - 42 + (100*n^2 - 320*n + 156)*x + (125*n^3 - 600*n^2 + 835*n - 312)*x^2)*P[n - 2, x] - ((50*n - 30)*x + (125*n^2 - 325*n + 150)*x^2 - (125*n^2 - 275*n + 120)*x^3)*D[P[n - 2, x], x] + (45*n - 62 + (50*n^2 - 160*n + 128)*x)*P[n - 1, x] + ((25*n - 40)*x - (25*n - 40)*x^2)*D[P[n - 1, x], x])/(5*n - 8)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n - 42 + (100*n^2 - 320*n + 156)*x + (125*n^3 - 600*n^2 + 835*n - 312)*x^2)*v[i - 2] - ((50*n - 30)*x + (125*n^2 - 325*n + 150)*x^2 - (125*n^2 - 275*n + 120)*x^3)*deriv(v[i - 2], x) + (45*n - 62 + (50*n^2 - 160*n + 128)*x)*v[i - 1] + ((25*n - 40)*x - (25*n - 40)*x^2)*deriv(v[i - 1], x))/(5*n - 8)); 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.