Real-rooted atlas

A102413 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(3 - 5\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(20 + 21\,x + 3\,x^{2} - 8\,n - 8\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(8 + 16\,x + 9\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
266113
314219145
4346846121161
58220519080151573
61985946994081231812041
74781673238717857491752117269
811544616774070643814124023624125889
92786125372415626040172267218190830627192205
1067263363073205910407137036716125102780385301328393
111623889309216733305481277046169246708622028438834733311169589
123920223521262961099194010219837236243601961267203121552445703614165553
13946426151731800474313570436184332915523168412470478221331746059689067639114835837
14228486159940250809439693432123864931119508673675913580604128935134189465653884879142152838617

Other recurrences

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

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 6 + 6\,x + x^{2}\)
Technical details
Node id
n-50746bd627deaa2a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-652a7cfa737886da)
Fingerprint
f6-65f5fe102fc0d40e

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:    2,     1;
  2:    6,     6,     1;
  3:   14,    21,     9,     1;
  4:   34,    68,    46,    12,     1;
  5:   82,   205,   190,    80,    15,    1;
  6:  198,   594,   699,   408,   123,   18,    1;
  7:  478,  1673,  2387,  1785,   749,  175,   21,   1;
  8: 1154,  4616,  7740,  7064,  3814, 1240,  236,  24,  1;
  9: 2786, 12537, 24156, 26040, 17226, 7218, 1908, 306, 27, 1;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n^2-5*n+3)*P(n-1,x) - (8*n-20+(8*n-21)*x+(n-3)*x^2)*d/dx P(n-1,x) + (8+16*x+9*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3)*P[n - 1, x] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*D[P[n - 1, x], x] + (8 + 16*x + 9*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 3)*v[i - 1] - (8*n - 20 + (8*n - 21)*x + (n - 3)*x^2)*deriv(v[i - 1], x) + (8 + 16*x + 9*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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