Real-rooted atlas

A257618 → D

nonnegative coefficients

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

Recurrence

\(\left(-7 + 4\,n\right)\,P_{n}(x) = \left(60 - 264\,x + 924\,x^{2} - 80\,n + 448\,n x - 2096\,n x^{2} - 128\,n^{2} x + 1344\,n^{2} x^{2} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48\,x - 384\,x^{2} + 336\,x^{3} - 64\,n x + 704\,n x^{2} - 640\,n x^{3} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-76 + 196\,x + 48\,n - 224\,n x + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-56\,x + 56\,x^{2} + 32\,n x - 32\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
240848
3472944241440
4492833984147846453760
54982486796813019521992961602419200
64995841901145667701504380229122497920384127733760
7499904038417126427298233603639764480960428160299942408967749181440

All 15 rows

Technical details
Node id
n-d8183db5bbaf8975 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6bc0895ac2a350b9

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:        40,            8;
  3:       472,          944,            24;
  4:      4928,        33984,         14784,             64;
  5:     49824,       867968,       1301952,         199296,            160;
  6:    499584,     19011456,      67701504,       38022912,        2497920,           384;
  7:   4999040,    384171264,    2729823360,     3639764480,      960428160,      29994240,          896;
  8:  49997824,   7414986752,   95178196992,   247503984640,   158630328320,   22244960256,    349984768,       2048;
  9: 499995136, 139269509120, 3030757128192, 13745114509312, 17181393136640, 6061514256384, 487443281920, 3999961088, 4608;
%F Formula
The row polynomials satisfy (4*n-7)*P(n,x) = -(80*n-60+(128*n^2-448*n+264)*x+(256*n^3-1344*n^2+2096*n-924)*x^2)*P(n-2,x) - ((64*n-48)*x+(256*n^2-704*n+384)*x^2-(256*n^2-640*n+336)*x^3)*d/dx P(n-2,x) + (48*n-76+(64*n^2-224*n+196)*x)*P(n-1,x) + ((32*n-56)*x-(32*n-56)*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[(-(80*n - 60 + (128*n^2 - 448*n + 264)*x + (256*n^3 - 1344*n^2 + 2096*n - 924)*x^2)*P[n - 2, x] - ((64*n - 48)*x + (256*n^2 - 704*n + 384)*x^2 - (256*n^2 - 640*n + 336)*x^3)*D[P[n - 2, x], x] + (48*n - 76 + (64*n^2 - 224*n + 196)*x)*P[n - 1, x] + ((32*n - 56)*x - (32*n - 56)*x^2)*D[P[n - 1, x], x])/(4*n - 7)]];
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] = (-(80*n - 60 + (128*n^2 - 448*n + 264)*x + (256*n^3 - 1344*n^2 + 2096*n - 924)*x^2)*v[i - 2] - ((64*n - 48)*x + (256*n^2 - 704*n + 384)*x^2 - (256*n^2 - 640*n + 336)*x^3)*deriv(v[i - 2], x) + (48*n - 76 + (64*n^2 - 224*n + 196)*x)*v[i - 1] + ((32*n - 56)*x - (32*n - 56)*x^2)*deriv(v[i - 1], x))/(4*n - 7)); 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.