Real-rooted atlas

A156365 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12\,x^{2} - 16\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x^{2} + 12\,x^{3} - 2\,n x^{2} + 4\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 + 8\,x + 4\,n x\right)\,P_{n-1}(x) + \left(x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 16\,x + 12\,x^{2}\)
\(P_{1}(x) = 1 + 44\,x + 132\,x^{2} + 32\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01161229
114413232209
21104792832801809
3122836249664456019218269
4148014292773129528023040448210853
5198851516499808124952082425611065610242737769
612008175296282348812495200169409286544384514048230439497657
714052574080145661441048283202515879682039260164898816023339525120626813813

All 15 rows

Technical details
Node id
n-81e7fb94c47d41ed · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f712c08c0fc317d2

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,    16,      12;
  1: 1,    44,     132,        32;
  2: 1,   104,     792,       832,         80;
  3: 1,   228,    3624,      9664,       4560,         192;
  4: 1,   480,   14292,     77312,      95280,       23040,         448;
  5: 1,   988,   51516,    499808,    1249520,      824256,      110656,        1024;
  6: 1,  2008,  175296,   2823488,   12495200,    16940928,     6544384,      514048,        2304;
  7: 1,  4052,  574080,  14566144,  104828320,   251587968,   203926016,    48988160,     2333952,       5120;
  8: 1,  8144, 1831644,  70511616,  779049120,  3019055616,  4362675072,  2256371712,   351675648,   10424320,    11264;
  9: 1, 16332, 5739252, 326005920, 5305477920, 31202358912, 72805504128, 67910117376, 23472426240, 2448747520, 45990912, 24576;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2+16*n+12)*x^2*P(n-2,x) - ((2*n+6)*x^2-(4*n+12)*x^3)*d/dx P(n-2,x) + (1+(4*n+8)*x)*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 16*x + 12*x^2, P(1,x) = 1 + 44*x + 132*x^2 + 32*x^3.
%t Mathematica
P[0, x_] := 1 + 16*x + 12*x^2;
P[1, x_] := 1 + 44*x + 132*x^2 + 32*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 + 16*n + 12)*x^2*P[n - 2, x] - ((2*n + 6)*x^2 - (4*n + 12)*x^3)*D[P[n - 2, x], x] + (1 + (4*n + 8)*x)*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 12*x^2; v[2] = 1 + 44*x + 132*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 + 16*n + 12)*x^2*v[i - 2] - ((2*n + 6)*x^2 - (4*n + 12)*x^3)*deriv(v[i - 2], x) + (1 + (4*n + 8)*x)*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], x)); 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.