Real-rooted atlas

A059473 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(12 + 4\,x - 9\,n - 2\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(74 + 52\,x + 8\,x^{2} - 26\,n - 16\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(52 + 58\,x + 20\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
24026470
3264256808608
4174422481060216165284
511520185281168836085443245920
6760961466881159204800810976131264399064
7502656112947210727045576961712643104030721283468032

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(4 + 2\,x\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)
Technical details
Node id
n-b35d6f3300b0ca76 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-b1b3df86c56d48cf)
Fingerprint
f6-b3ff1427b4bfa2c0

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,        2;
  2:       40,       26,        4;
  3:      264,      256,       80,        8;
  4:     1744,     2248,     1060,      216,       16;
  5:    11520,    18528,    11688,     3608,      544,      32;
  6:    76096,   146688,   115920,    48008,    10976,    1312,      64;
  7:   502656,  1129472,  1072704,   557696,   171264,   31040,    3072,    128;
  8:  3320320,  8521088,  9452224,  5915456,  2282896,  555968,   83392,   7040,   256;
  9: 21932544, 63290368, 80305280, 58773376, 27328736, 8368288, 1686656, 215680, 15872, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n^2-9*n+12-(2*n-4)*x)*P(n-1,x) - (26*n-74+(16*n-52)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (52+58*x+20*x^2+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[((3*n^2 - 9*n + 12 - (2*n - 4)*x)*P[n - 1, x] - (26*n - 74 + (16*n - 52)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (52 + 58*x + 20*x^2 + 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] = ((3*n^2 - 9*n + 12 - (2*n - 4)*x)*v[i - 1] - (26*n - 74 + (16*n - 52)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (52 + 58*x + 20*x^2 + 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