Real-rooted atlas

A100861 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(3\right)\,P_{n}(x) = \left(-9 - 9\,x + 12\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-32 - 48\,x - 18\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2538
313922
4739027190
5281390135806
61741328520254057456
784851770312285283541308
857233146244137970567008505406652
9328753916380955962442260765452719900
102389141772645598714706208650191362522963528338976
1115539261542442457560135052577910182432252525985218731976
121206614654730499187649190456515170203073281757577955075779552400833128
1386654599336361870746346576665589690530030758077358537829309851341520774319112
14714094217332940579945645713138076960763309544466036615167801183553447969525295540245239250133760

Other recurrences

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

\(P_{n}(x) = \left(-4 - 3\,x + 4\,n + 3\,n x\right)\,P_{n-2}(x) + \left(1\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
Technical details
Node id
n-62c21d7097227e9f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ba776593a5d483a1)
Fingerprint
f6-ca873dd047c5050a

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:      5,      3;
  3:     13,      9;
  4:     73,     90,     27;
  5:    281,    390,    135;
  6:   1741,   3285,   2025,    405;
  7:   8485,  17703,  12285,   2835;
  8:  57233, 146244, 137970,  56700,  8505;
  9: 328753, 916380, 955962, 442260, 76545;
%F Formula
The row polynomials satisfy 3*P(n,x) = (12*n-9+(9*n-9)*x)*P(n-1,x) - (32+48*x+18*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[((12*n - 9 + (9*n - 9)*x)*P[n - 1, x] - (32 + 48*x + 18*x^2)*D[P[n - 1, x], x])/(3)]];
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] = ((12*n - 9 + (9*n - 9)*x)*v[i - 1] - (32 + 48*x + 18*x^2)*deriv(v[i - 1], x))/(3)); 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