Real-rooted atlas

A257612 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(12 - 40\,x + 60\,x^{2} - 24\,n + 96\,n x - 184\,n x^{2} - 32\,n^{2} x + 144\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,x - 32\,x^{2} + 24\,x^{3} - 16\,n x + 80\,n x^{2} - 64\,n x^{3} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-20 + 36\,x + 16\,n - 48\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 12\,x^{2} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
224832
318436824576
41216736036486412288
5758410764816147230336160307200
6462081349504452121626990082310403848847360
72789121552819299733632132978176388204801673472896289013760

All 15 rows

Technical details
Node id
n-9da9c144cb7ef765 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a7d11e539b3a8882

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:       24,          8;
  3:      184,        368,          24;
  4:     1216,       7360,        3648,           64;
  5:     7584,     107648,      161472,        30336,          160;
  6:    46208,    1349504,     4521216,      2699008,       231040,         384;
  7:   278912,   15528192,    99733632,    132978176,     38820480,     1673472,        896;
  8:  1677312,  169785344,  1908701184,   4787214336,   3181168640,   509356032,   11741184,     2048;
  9: 10072576, 1798492160, 33343444992, 142158426112, 177698032640, 66686889984, 6294722560, 80580608, 4608;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(24*n-12+(32*n^2-96*n+40)*x+(32*n^3-144*n^2+184*n-60)*x^2)*P(n-2,x) - ((16*n-8)*x+(32*n^2-80*n+32)*x^2-(32*n^2-64*n+24)*x^3)*d/dx P(n-2,x) + (16*n-20+(16*n^2-48*n+36)*x)*P(n-1,x) + ((8*n-12)*x-(8*n-12)*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[(-(24*n - 12 + (32*n^2 - 96*n + 40)*x + (32*n^3 - 144*n^2 + 184*n - 60)*x^2)*P[n - 2, x] - ((16*n - 8)*x + (32*n^2 - 80*n + 32)*x^2 - (32*n^2 - 64*n + 24)*x^3)*D[P[n - 2, x], x] + (16*n - 20 + (16*n^2 - 48*n + 36)*x)*P[n - 1, x] + ((8*n - 12)*x - (8*n - 12)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = (-(24*n - 12 + (32*n^2 - 96*n + 40)*x + (32*n^3 - 144*n^2 + 184*n - 60)*x^2)*v[i - 2] - ((16*n - 8)*x + (32*n^2 - 80*n + 32)*x^2 - (32*n^2 - 64*n + 24)*x^3)*deriv(v[i - 2], x) + (16*n - 20 + (16*n^2 - 48*n + 36)*x)*v[i - 1] + ((8*n - 12)*x - (8*n - 12)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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.