Real-rooted atlas

A145324 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2617629
310619824193
4151565336341201458
52133018954756464472012366
62861752902208145244382685040116568
73610571258677721262724464444351504403201210392

All 15 rows

Technical details
Node id
n-68f9f1a41a229834 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d4a70c002cd58c1f

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:  3,    2;
  2:  6,   17,     6;
  3: 10,   61,    98,     24;
  4: 15,  156,   533,    634,     120;
  5: 21,  330,  1895,   4756,    4644,      720;
  6: 28,  617,  5290,  22081,   45244,    38268,     5040;
  7: 36, 1057, 12586,  77721,  262724,   464444,   351504,    40320;
  8: 45, 1696, 26698, 227220, 1128029,  3268700,  5154876,  3566736,   362880;
  9: 55, 2586, 51918, 580440, 3946719, 16695750, 42931916, 61752456, 39659040, 3628800;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];
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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1])/(n)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.