Real-rooted atlas

A156992 → basis A008277 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6\,n - 4\,n x - n x^{2} - 6\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,n x - n x^{2} - 2\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(5 + 2\,x + 5\,n + 2\,n x\right)\,P_{n-1}(x) + \left(x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2301242
345643272960
4780013200504048026520
5151920410400270000576003600893520
63351600136180801375920049392006804003024036378720
7830188804894041607137849603838464008749440084672002822401766298240

All 15 rows

Technical details
Node id
n-6363fc33189ae9e5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-306ed4af19c47352

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           0;
  1:           2;
  2:          30,           12;
  3:         456,          432,            72;
  4:        7800,        13200,          5040,           480;
  5:      151920,       410400,        270000,         57600,          3600;
  6:     3351600,     13618080,      13759200,       4939200,        680400,        30240;
  7:    83018880,    489404160,     713784960,     383846400,      87494400,      8467200,      282240;
  8:  2287958400,  19112889600,   38727279360,   29293125120,    9830419200,   1554577920,   111767040,    2903040;
  9: 69567724800, 810274752000, 2223057312000, 2273951232000, 1054293408000, 242331264000, 28195776000, 1567641600, 32659200;
%F Formula
The row polynomials satisfy P(n,x) = -(6*n^2+6*n+(4*n^2+4*n)*x+(n^2+n)*x^2)*P(n-2,x) - ((2*n^2+2*n)*x+(n^2+n)*x^2)*d/dx P(n-2,x) + (5*n+5+(2*n+2)*x)*P(n-1,x) + (n+1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(6*n^2 + 6*n + (4*n^2 + 4*n)*x + (n^2 + n)*x^2)*P[n - 2, x] - ((2*n^2 + 2*n)*x + (n^2 + n)*x^2)*D[P[n - 2, x], x] + (5*n + 5 + (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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(6*n^2 + 6*n + (4*n^2 + 4*n)*x + (n^2 + n)*x^2)*v[i - 2] - ((2*n^2 + 2*n)*x + (n^2 + n)*x^2)*deriv(v[i - 2], x) + (5*n + 5 + (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.