Real-rooted atlas

A286718 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 48\,x + 81\,x^{2} + 24\,n - 36\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 36\,x^{3} + 36\,n x^{2}\right)\,P_{n-2}'(x) + \left(-5 + 12\,x + 6\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121214
2108481175
38079212154322519
488096721871111232202542520
512320145968327240257040810008748832316
620944026292966548364611496025818754986363572118618292
741888005509920014871502815722380880301375211264202750517139968469545116

All 15 rows

Technical details
Node id
n-a5bad3663216dcf6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e51ce4670dc45695

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:          2,          12;
  2:         10,          84,           81;
  3:         80,         792,         1215,          432;
  4:        880,        9672,        18711,        11232,        2025;
  5:      12320,      145968,       327240,       257040,       81000,        8748;
  6:     209440,     2629296,      6548364,      6114960,     2581875,      498636,      35721;
  7:    4188800,    55099200,    148715028,    157223808,    80301375,    21126420,    2750517,    139968;
  8:   96342400,  1317547200,   3792365244,   4409294400,  2583918225,   832809600,  149528106,  13996800,   531441;
  9: 2504902400, 35412336000, 107494939944, 134867602368, 87850441350, 32815576332, 7288369956, 949822848, 66961566, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(36*n-48)*x-81*x^2)*P(n-2,x) + ((36*n-48)*x^2+36*x^3)*d/dx P(n-2,x) + (6*n-5+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (36*n - 48)*x - 81*x^2)*P[n - 2, x] + ((36*n - 48)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (6*n - 5 + 12*x)*P[n - 1, x] - 12*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; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (36*n - 48)*x - 81*x^2)*v[i - 2] + ((36*n - 48)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (6*n - 5 + 12*x)*v[i - 1] - 12*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

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.