Real-rooted atlas

A081580 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15 + 2\,x - 18\,n - n x + 6\,n^{2}\right)\,P_{n-1}(x) + \left(127 + 71\,x + 4\,x^{2} - 49\,n - 26\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(98 + 101\,x + 28\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21911131
38777191184
44515231992711201
52223322516953853517564
6111791938312915411563543148271
75576711269391203375858295949511306544

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(10 + 5\,x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-24757ab977a41ca8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-a13317a426b24d26)
Fingerprint
f6-d2d93ab8aeb89acd

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:      19,      11,       1;
  3:      87,      77,      19,       1;
  4:     451,     523,     199,      27,      1;
  5:    2223,    3225,    1695,     385,     35,      1;
  6:   11179,   19383,   12915,    4115,    635,     43,     1;
  7:   55767,  112693,   91203,   37585,   8295,    949,    51,    1;
  8:  279091,  643571,  612367,  309683,  89395,  14747,  1327,   59,  1;
  9: 1394943, 3615569, 3956167, 2373281, 848743, 184601, 23983, 1769, 67, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (6*n^2-18*n+15-(n-2)*x)*P(n-1,x) - (49*n-127+(26*n-71)*x+(n-4)*x^2)*d/dx P(n-1,x) + (98+101*x+28*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[Together[((6*n^2 - 18*n + 15 - (n - 2)*x)*P[n - 1, x] - (49*n - 127 + (26*n - 71)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (98 + 101*x + 28*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n)]];
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] = ((6*n^2 - 18*n + 15 - (n - 2)*x)*v[i - 1] - (49*n - 127 + (26*n - 71)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (98 + 101*x + 28*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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

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