Real-rooted atlas

A089503 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
33012143
4336168241529
5504028805404018501
69504059400132001320601169021
7216216014414403603604368027308414010455
8576576004036032011007360152880011760050401121110676833

All 15 rows

Technical details
Node id
n-2e9b77d16b34b95b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fe953b9def4ff3da

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:           1;
   2:           4,           1;
   3:          30,          12,           1;
   4:         336,         168,          24,          1;
   5:        5040,        2880,         540,         40,         1;
   6:       95040,       59400,       13200,       1320,        60,        1;
   7:     2162160,     1441440,      360360,      43680,      2730,       84,      1;
   8:    57657600,    40360320,    11007360,    1528800,    117600,     5040,    112,     1;
   9:  1764322560,  1283143680,   374250240,   57576960,   5140800,   274176,   8568,   144,   1;
  10: 60949324800, 45711993600, 14065228800, 2344204800, 234420480, 14651280, 574560, 13680, 180, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-2+x)*P(n-1,x) + (n-1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n - 2 + x)*P[n - 1, x] + (n - 1 + x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.