Real-rooted atlas

A257622 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,n\right)\,P_{n}(x) = \left(-16 + 16\,x + 8\,x^{2} - 12\,n - 12\,n x + 18\,n x^{2} - 18\,n^{2} x - 27\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 12\,x^{3} - 9\,n x - 36\,n x^{2} + 45\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(8 + 2\,x + 15\,n + 12\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
21611248176
364110416562563080
425693923312018784128062832
510247393651434468579218484061441466080
640965564646917904178305921152984016693922867238536960
716384407547285038264379379648474224560170076528142641521310721127206080

All 15 rows

Technical details
Node id
n-bc47a5f111fa8635 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1e1f504198d7902a

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,       112,          48;
  3:     64,      1104,        1656,          256;
  4:    256,      9392,       33120,        18784,         1280;
  5:   1024,     73936,      514344,       685792,       184840,         6144;
  6:   4096,    556464,     6917904,     17830592,     11529840,      1669392,        28672;
  7:  16384,   4075472,    85038264,    379379648,    474224560,    170076528,     14264152,      131072;
  8:  65536,  29347504,   984873216,   7086238112,  15175185920,  10629357168,   2298037504,   117390016,    589824;
  9: 262144, 209102544, 10949263560, 121010709792, 411101129880, 493321355856, 211768742136, 29198036160, 940961448, 2621440;
%F Formula
The row polynomials satisfy (3*n+1)*P(n,x) = -(12*n+16+(18*n^2+12*n-16)*x+(27*n^3+27*n^2-18*n-8)*x^2)*P(n-2,x) - ((9*n+12)*x+(27*n^2+36*n)*x^2-(27*n^2+45*n+12)*x^3)*d/dx P(n-2,x) + (15*n+8+(18*n^2+12*n+2)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*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[(-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*P[n - 2, x] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*D[P[n - 2, x], x] + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x])/(3*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] = (-(12*n + 16 + (18*n^2 + 12*n - 16)*x + (27*n^3 + 27*n^2 - 18*n - 8)*x^2)*v[i - 2] - ((9*n + 12)*x + (27*n^2 + 36*n)*x^2 - (27*n^2 + 45*n + 12)*x^3)*deriv(v[i - 2], x) + (15*n + 8 + (18*n^2 + 12*n + 2)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x))/(3*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

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.