Real-rooted atlas

A019538 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
31121831
412810896233
51604509606002071
61124162062409000432021305
71252541833600840009072035280249271
815081738816329663000011491209878403225603270713

All 15 rows

Technical details
Node id
n-b9e1bcc0efba62aa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ea3a19fb0ca3a7eb

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,    4;
   3: 1,   12,     18;
   4: 1,   28,    108,      96;
   5: 1,   60,    450,     960,      600;
   6: 1,  124,   1620,    6240,     9000,     4320;
   7: 1,  252,   5418,   33600,    84000,    90720,     35280;
   8: 1,  508,  17388,  163296,   630000,  1149120,    987840,    322560;
   9: 1, 1020,  54450,  745920,  4170600, 11430720,  16299360,  11612160,   3265920;
  10: 1, 2044, 167940, 3274080, 25515000, 98612640, 207446400, 241920000, 146966400, 36288000;
%F Formula
The row polynomials satisfy P(n,x) = -2*x^2*P(n-2,x) - (x^2+x^3)*d/dx P(n-2,x) + (1+4*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-2*x^2*P[n - 2, x] - (x^2 + x^3)*D[P[n - 2, x], x] + (1 + 4*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = -2*x^2*v[i - 2] - (x^2 + x^3)*deriv(v[i - 2], x) + (1 + 4*x)*v[i - 1] + (x + 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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.