Real-rooted atlas

A257613 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-16 - 8\,n - 16\,n x - 8\,n x^{2} - 8\,n^{2} x - 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x + 8\,x^{3} - 4\,n x - 8\,n x^{2} + 12\,n x^{3} - 4\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(8 + 4\,x + 6\,n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2169648160
36483212482562400
425662721996812544128040320
51024437762538243384321094406144752640
640962913282818560676864046976008739842867215482880
716384187904028666368112783360140979200573327366576640131072348364800

All 15 rows

Technical details
Node id
n-db27be2f5676b05b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7c3b635378939853

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:      1;
  1:      4,        8;
  2:     16,       96,         48;
  3:     64,      832,       1248,         256;
  4:    256,     6272,      19968,       12544,        1280;
  5:   1024,    43776,     253824,      338432,      109440,        6144;
  6:   4096,   291328,    2818560,     6768640,     4697600,      873984,       28672;
  7:  16384,  1879040,   28666368,   112783360,   140979200,    57332736,     6576640,     131072;
  8:  65536, 11864064,  274427904,  1662939136,  3383500800,  2494408704,   640331776,   47456256,    589824;
  9: 262144, 73805824, 2515752960, 22483853312, 69703444480, 83644133376, 39346743296, 6708674560, 332126208, 2621440;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(8*n+16+(8*n^2+16*n)*x+(4*n^3+12*n^2+8*n)*x^2)*P(n-2,x) - ((4*n+8)*x+(4*n^2+8*n)*x^2-(4*n^2+12*n+8)*x^3)*d/dx P(n-2,x) + (6*n+8+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*P[n - 2, x] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*D[P[n - 2, x], x] + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n + 16 + (8*n^2 + 16*n)*x + (4*n^3 + 12*n^2 + 8*n)*x^2)*v[i - 2] - ((4*n + 8)*x + (4*n^2 + 8*n)*x^2 - (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 2], x) + (6*n + 8 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.