Real-rooted atlas

A128813 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
21220840
314432423656760
4288080648284371661623560
5864002880003775442440247793698561083760
636288001399680022192848185549768641840212854421683269360640
72032128008890560001648706688168267124810220373443698118247387036862881285895654400

All 15 rows

Technical details
Node id
n-96014d301579d4a7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-bd198274b898c9ae

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:             1;
  1:             2,             2;
  2:            12,            20,              8;
  3:           144,           324,            236,             56;
  4:          2880,          8064,           8284,           3716,            616;
  5:         86400,        288000,         377544,         244024,          77936,           9856;
  6:       3628800,      13996800,       22192848,       18554976,        8641840,        2128544,        216832;
  7:     203212800,     889056000,     1648706688,     1682671248,     1022037344,      369811824,      73870368,       6288128;
  8:   14631321600,   71530905600,   151601953536,   182154477312,   135845524944,    64441833056,   19001703984,    3185948832,    232660736;
  9: 1316818944000, 7110822297600, 16934597475840, 23367592820736, 20605203201312, 12048659122464, 4674477679136, 1160813778144, 167493112512, 10702393856;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2*n^2+2*n+(n^2+n+2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 2*n + (n^2 + n + 2)*x)*P[n - 1, x])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + (n^2 + n + 2)*x)*v[i - 1])/(2)); 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.