Real-rooted atlas

A081579 → x ↦ x+1 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(13 + 2\,x - 15\,n - n x + 5\,n^{2}\right)\,P_{n-1}(x) + \left(107 + 61\,x + 4\,x^{2} - 41\,n - 22\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(82 + 85\,x + 24\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21710128
37567171160
4361424166241976
51683246913263143115824
67937139269471312451138135008
7372757619962823266596161757451209920

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(8 + 4\,x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-d60414e3ab73b5cf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-93db93aa6db882a4)
Fingerprint
f6-b116ee9a41c824cb

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:     17,      10,       1;
  3:     75,      67,      17,       1;
  4:    361,     424,     166,      24,      1;
  5:   1683,    2469,    1326,     314,     31,      1;
  6:   7937,   13926,    9471,    3124,    511,     38,     1;
  7:  37275,   76199,   62823,   26659,   6161,    757,    45,    1;
  8: 175321,  409028,  396140,  205676,  61726,  10780,  1052,   52,  1;
  9: 824163, 2161097, 2404828, 1477732, 546778, 124766, 17324, 1396, 59, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (5*n^2-15*n+13-(n-2)*x)*P(n-1,x) - (41*n-107+(22*n-61)*x+(n-4)*x^2)*d/dx P(n-1,x) + (82+85*x+24*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[((5*n^2 - 15*n + 13 - (n - 2)*x)*P[n - 1, x] - (41*n - 107 + (22*n - 61)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (82 + 85*x + 24*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] = ((5*n^2 - 15*n + 13 - (n - 2)*x)*v[i - 1] - (41*n - 107 + (22*n - 61)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (82 + 85*x + 24*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