Real-rooted atlas

A143497 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + x + n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 30 + 12\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03012143
1210126211358
2168013443363213393
31512015120504072045136046
415120018144075600144001350601424051
51663200232848011642402772003465023107715470158
6199584003193344018627840532224083160073920369696176751233
7259459200467026560311351040103783680194594402162160144144561611711163391958

All 15 rows

Technical details
Node id
n-1d88497fc221d044 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1d70f6b910bc9524

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          30,           12,           1;
  1:         210,          126,          21,           1;
  2:        1680,         1344,         336,          32,           1;
  3:       15120,        15120,        5040,         720,          45,          1;
  4:      151200,       181440,       75600,       14400,        1350,         60,         1;
  5:     1663200,      2328480,     1164240,      277200,       34650,       2310,        77,        1;
  6:    19958400,     31933440,    18627840,     5322240,      831600,      73920,      3696,       96,      1;
  7:   259459200,    467026560,   311351040,   103783680,    19459440,    2162160,    144144,     5616,    117,     1;
  8:  3632428800,   7264857600,  5448643200,  2075673600,   454053600,   60540480,   5045040,   262080,   8190,   140,   1;
  9: 54486432000, 119870150400, 99891792000, 42810768000, 10702692000, 1664863200, 166486320, 10810800, 450450, 11550, 165, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+6+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 30 + 12*x + x^2.
%t Mathematica
P[0, x_] := 30 + 12*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 6 + x)*P[n - 1, x] + 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] = 30 + 12*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 6 + x)*v[i - 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.