Real-rooted atlas

A156992 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,n - 2\,n x - n x^{2} - 2\,n^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-n x - n x^{2} - n^{2} x - n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(3 + 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x) + \left(x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
2246018102
3192912648961848
419201560019800672060044640
5230403038406156003600007200043201378800
63225606703200204271201834560061740008164803528052824240
7516096016603776073410624095171328047980800010499328098784003225602452020480

All 15 rows

Technical details
Node id
n-4a6fbeb2444d3f2e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9be755efdd908b94

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:         24,           60,            18;
  3:        192,          912,           648,            96;
  4:       1920,        15600,         19800,          6720,           600;
  5:      23040,       303840,        615600,        360000,         72000,          4320;
  6:     322560,      6703200,      20427120,      18345600,       6174000,        816480,        35280;
  7:    5160960,    166037760,     734106240,     951713280,     479808000,     104993280,      9878400,      322560;
  8:   92897280,   4575916800,   28669334400,   51636372480,   36616406400,   11796503040,   1813674240,   127733760,    3265920;
  9: 1857945600, 139135449600, 1215412128000, 2964076416000, 2842439040000, 1265152089600, 282719808000, 32223744000, 1763596800, 36288000;
%F Formula
The row polynomials satisfy P(n,x) = -(2*n^2+2*n+(2*n^2+2*n)*x+(n^2+n)*x^2)*P(n-2,x) - ((n^2+n)*x+(n^2+n)*x^2)*d/dx P(n-2,x) + (3*n+3+(2*n+2)*x)*P(n-1,x) + (n+1)*x*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[-(2*n^2 + 2*n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*P[n - 2, x] - ((n^2 + n)*x + (n^2 + n)*x^2)*D[P[n - 2, x], x] + (3*n + 3 + (2*n + 2)*x)*P[n - 1, x] + (n + 1)*x*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] = -(2*n^2 + 2*n + (2*n^2 + 2*n)*x + (n^2 + n)*x^2)*v[i - 2] - ((n^2 + n)*x + (n^2 + n)*x^2)*deriv(v[i - 2], x) + (3*n + 3 + (2*n + 2)*x)*v[i - 1] + (n + 1)*x*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. [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.