Real-rooted atlas

A219512 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21512128
3105141331280
494518307867813640
51039526685175383450171158240
61351354359603964951253521325736011106560
7202702579213059356175423881175329146971741124344320

All 15 rows

Technical details
Node id
n-817edc0b1e94c6f2 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-0a12f7fa088d0629

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:         3,          1;
  2:        15,         12,          1;
  3:       105,        141,         33,          1;
  4:       945,       1830,        786,         78,          1;
  5:     10395,      26685,      17538,       3450,        171,         1;
  6:    135135,     435960,     396495,     125352,      13257,       360,        1;
  7:   2027025,    7921305,    9356175,    4238811,     753291,     46971,      741,      1;
  8:  34459425,  158799690,  233216460,  140913270,   37013166,   4046526,   157956,   1506,    1;
  9: 654729075, 3486128625, 6167943180, 4732607160, 1696782438, 282182454, 20135004, 513024, 3039, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+1+n*x)*P(n-1,x) + (x-x^2)*d/dx 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 + 1 + n*x)*P[n - 1, x] + (x - x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + n*x)*v[i - 1] + (x - x^2)*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. [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