Real-rooted atlas

A304236 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-36 - 24\,x + 38\,n + 24\,n x\right)\,P_{n-1}(x) + \left(-111 - 146\,x - 48\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
210616
3191231
410912636271
5280348108736
61261213611882164801
737816948424886415841
81513033738277569936129687856
949159118956107676432006480325471
101853295133785599082991607776077761643311
11627760187893622427281334016395280466566525376
122295721761131210362168738590428900805832004665631175041
137945561282882964182033632848416144516963374784326592129055681
142860705011056443018074772016149456084777840259640644245696279936596681296

Other recurrences

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

\(P_{n}(x) = \left(9 + 6\,x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-f050517ea751864e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-a0db61b1ca32d092)
Fingerprint
f6-b5cdec76cd14acc3

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:     1;
  2:    10,      6;
  3:    19,     12;
  4:   109,    126,     36;
  5:   280,    348,    108;
  6:  1261,   2136,   1188,   216;
  7:  3781,   6948,   4248,   864;
  8: 15130,  33738,  27756,  9936, 1296;
  9: 49159, 118956, 107676, 43200, 6480;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (38*n-36+(24*n-24)*x)*P(n-1,x) - (111+146*x+48*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[Together[((38*n - 36 + (24*n - 24)*x)*P[n - 1, x] - (111 + 146*x + 48*x^2)*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] = ((38*n - 36 + (24*n - 24)*x)*v[i - 1] - (111 + 146*x + 48*x^2)*deriv(v[i - 1], x))/(2*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