Real-rooted atlas

A019538 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-8\,x + 22\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 26\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3} + 5\,x^{4}\right)\,P_{n-2}''(x) + \left(1 + 8\,x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2189
31245479
4156324384765
511201350384030008311
612484860249604500025920100989
71504162541344004200005443202469601362439
81101652164653184315000068947206914880258048020246445

All 15 rows

Technical details
Node id
n-be6f49875ee2b6af · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c70146f291298cbd

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,    8;
   3: 1,   24,     54;
   4: 1,   56,    324,      384;
   5: 1,  120,   1350,     3840,      3000;
   6: 1,  248,   4860,    24960,     45000,     25920;
   7: 1,  504,  16254,   134400,    420000,    544320,     246960;
   8: 1, 1016,  52164,   653184,   3150000,   6894720,    6914880,    2580480;
   9: 1, 2040, 163350,  2983680,  20853000,  68584320,  114095520,   92897280,   29393280;
  10: 1, 4088, 503820, 13096320, 127575000, 591675840, 1452124800, 1935360000, 1322697600, 362880000;
%F Formula
The row polynomials satisfy P(n,x) = -(8*x-22*x^2)*P(n-2,x) - (2*x-26*x^3)*d/dx P(n-2,x) - (x^2-4*x^3-5*x^4)*d^2/dx^2 P(n-2,x) + (1+8*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(8*x - 22*x^2)*P[n - 2, x] - (2*x - 26*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3 - 5*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 8*x)*P[n - 1, x] + (2*x - 4*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 + 8*x; for(n = 3, N, my(i = n); v[i] = -(8*x - 22*x^2)*v[i - 2] - (2*x - 26*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3 - 5*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 8*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, v[i])))};
T(10)

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.