Real-rooted atlas

A120910 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(9 + 24\,x + 16\,x^{2} + 12\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1211233
29916848315
3405118810081922793
4153964809504537676823667
55589307806480063360268803072194481
619683134136369360518400380160129024122881563051
76779755112418779043447360362880021288966021124915212353145

All 15 rows

Technical details
Node id
n-2024ae9472770903 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-62f46345657f9dbc

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      3;
  1:     21,      12;
  2:     99,     168,       48;
  3:    405,    1188,     1008,       192;
  4:   1539,    6480,     9504,      5376,       768;
  5:   5589,   30780,    64800,     63360,     26880,      3072;
  6:  19683,  134136,   369360,    518400,    380160,    129024,     12288;
  7:  67797,  551124,  1877904,   3447360,   3628800,   2128896,    602112,    49152;
  8: 229635, 2169504,  8817984,  20030976,  27578880,  23224320,  11354112,  2752512,   196608;
  9: 767637, 8266860, 39051072, 105815808, 180278784, 198567936, 139345920, 58392576, 12386304, 786432;
%F Formula
The row polynomials satisfy (4*n-1+4*x)*P(n,x) = (12*n+9+(16*n+24)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 9 + (16*n + 24)*x + 16*x^2)*P[n - 1, x])/(4*n - 1 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 9 + (16*n + 24)*x + 16*x^2)*v[i - 1])/(4*n - 1 + 4*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.