Real-rooted atlas

A257616 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 3\,n\right)\,P_{n}(x) = \left(32 - 128\,x + 320\,x^{2} - 48\,n + 240\,n x - 792\,n x^{2} - 72\,n^{2} x + 540\,n^{2} x^{2} - 108\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(24\,x - 144\,x^{2} + 120\,x^{3} - 36\,n x + 288\,n x^{2} - 252\,n x^{3} - 108\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-44 + 100\,x + 30\,n - 120\,n x + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 30\,x^{2} + 18\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
232840
331262424960
426561747279686428160
52166435084852627286656160985600
61743366038400210508801207680087168038440212480
713971209569510465651520087535360023923776083827208961876582400

All 15 rows

Technical details
Node id
n-3f9e261fec4bd38f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d87433d07b949f42

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:       32,           8;
  3:      312,         624,           24;
  4:     2656,       17472,         7968,            64;
  5:    21664,      350848,       526272,         86656,           160;
  6:   174336,     6038400,     21050880,      12076800,        871680,          384;
  7:  1397120,    95695104,    656515200,     875353600,     239237760,      8382720,         896;
  8: 11182592,  1445912576,  17723668992,   45518387200,   29539448320,   4337737728,    78278144,      2048;
  9: 89473536, 21226844160, 436890396672, 1939687944192, 2424609930240, 873780793344, 74293954560, 715788288, 4608;
%F Formula
The row polynomials satisfy (3*n-5)*P(n,x) = -(48*n-32+(72*n^2-240*n+128)*x+(108*n^3-540*n^2+792*n-320)*x^2)*P(n-2,x) - ((36*n-24)*x+(108*n^2-288*n+144)*x^2-(108*n^2-252*n+120)*x^3)*d/dx P(n-2,x) + (30*n-44+(36*n^2-120*n+100)*x)*P(n-1,x) + ((18*n-30)*x-(18*n-30)*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[(-(48*n - 32 + (72*n^2 - 240*n + 128)*x + (108*n^3 - 540*n^2 + 792*n - 320)*x^2)*P[n - 2, x] - ((36*n - 24)*x + (108*n^2 - 288*n + 144)*x^2 - (108*n^2 - 252*n + 120)*x^3)*D[P[n - 2, x], x] + (30*n - 44 + (36*n^2 - 120*n + 100)*x)*P[n - 1, x] + ((18*n - 30)*x - (18*n - 30)*x^2)*D[P[n - 1, x], x])/(3*n - 5)]];
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] = (-(48*n - 32 + (72*n^2 - 240*n + 128)*x + (108*n^3 - 540*n^2 + 792*n - 320)*x^2)*v[i - 2] - ((36*n - 24)*x + (108*n^2 - 288*n + 144)*x^2 - (108*n^2 - 252*n + 120)*x^3)*deriv(v[i - 2], x) + (30*n - 44 + (36*n^2 - 120*n + 100)*x)*v[i - 1] + ((18*n - 30)*x - (18*n - 30)*x^2)*deriv(v[i - 1], x))/(3*n - 5)); 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.