Real-rooted atlas

A225471 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 + 16\,x + 9\,x^{2} + 36\,n - 16\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(8\,x - 16\,x^{2} + 4\,x^{3} - 8\,n x + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(-5 + 4\,x + 8\,n\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
23252993
3384804243161447
461441520461837362528292
512288034386017023527920175036666681
629491209088020519979510564809262535644918389653
782575360275349060176441895419949604614750250992651764581233598

All 15 rows

Technical details
Node id
n-a304052d835c12d3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ba526ebf2da93975

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,            4;
  2:          32,           52,            9;
  3:         384,          804,          243,          16;
  4:        6144,        15204,         6183,         736,          25;
  5:      122880,       343860,       170235,       27920,        1750,        36;
  6:     2949120,      9088020,      5199795,     1056480,       92625,      3564,       49;
  7:    82575360,    275349060,    176441895,    41994960,     4614750,    250992,     6517,      64;
  8:  2642411520,   9416820420,   6618559815,  1783498080,   231748125,  15931944,   589274,   11008,    81;
  9: 95126814720, 358992001620, 272693118915, 81322753680, 12056640750, 986927004, 46434654, 1243008, 17496, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-36*n+20+(16*n-16)*x-9*x^2)*P(n-2,x) - ((8*n-8)*x-(16*n-16)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (8*n-5+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 36*n + 20 + (16*n - 16)*x - 9*x^2)*P[n - 2, x] - ((8*n - 8)*x - (16*n - 16)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (8*n - 5 + 4*x)*P[n - 1, x] + (2*x - 4*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] = 4 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 36*n + 20 + (16*n - 16)*x - 9*x^2)*v[i - 2] - ((8*n - 8)*x - (16*n - 16)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (8*n - 5 + 4*x)*v[i - 1] + (2*x - 4*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.