Real-rooted atlas

A241171 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1617
290301121
32520126012613907
411340075600132305101202741
57484400623700015800401267202046115430207
6681080400681080400227026800288288001171170819011619195761
7817296480009535125600039502663200697296600049405356010663380327661224061282907

All 15 rows

Technical details
Node id
n-277dc4a4d0ef7c4f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-18f46c82cbc91532

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:                6,                1;
  2:               90,               30,                1;
  3:             2520,             1260,              126,               1;
  4:           113400,            75600,            13230,             510,              1;
  5:          7484400,          6237000,          1580040,          126720,           2046,             1;
  6:        681080400,        681080400,        227026800,        28828800,        1171170,          8190,            1;
  7:      81729648000,      95351256000,      39502663200,      6972966000,      494053560,      10663380,        32766,         1;
  8:   12504636144000,   16672848192000,    8266953895200,   1882311631200,   196556560200,    8203431600,     96461910,    131070,      1;
  9: 2375880867360000, 3563821301040000, 2059096751712000, 576371099304000, 80735951696400, 5283068427000, 133955839080, 870123240, 524286, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n^2+3*n+1+n^2*x)*P(n-1,x) - ((4*n+1)*x+(2*n-1)*x^2)*d/dx P(n-1,x) + (2*x^2+x^3)*d^2/dx^2 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[(2*n^2 + 3*n + 1 + n^2*x)*P[n - 1, x] - ((4*n + 1)*x + (2*n - 1)*x^2)*D[P[n - 1, x], x] + (2*x^2 + x^3)*D[P[n - 1, x], {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 + 3*n + 1 + n^2*x)*v[i - 1] - ((4*n + 1)*x + (2*n - 1)*x^2)*deriv(v[i - 1], x) + (2*x^2 + x^3)*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.