Real-rooted atlas

A178820 → basis A013611 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-72 + 48\,x - 60\,n + 40\,n x - 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-2}(x) + \left(-24 - 12\,x + 16\,n + 8\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 24 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1241640
23606401601160
34320131208320128027040
445360197120219520806408960551600
54354562469376408576025804806594565734410287872
63919104274391046130790457372672246865924816896344064179886336
73359232027995136079472640010097049606372556802059468803244032019660802995584000

All 15 rows

Technical details
Node id
n-854160786720f84e · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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:         24,          16;
  2:        360,         640,         160;
  3:       4320,       13120,        8320,         1280;
  4:      45360,      197120,      219520,        80640,         8960;
  5:     435456,     2469376,     4085760,      2580480,       659456,        57344;
  6:    3919104,    27439104,    61307904,     57372672,     24686592,      4816896,      344064;
  7:   33592320,   279951360,   794726400,   1009704960,    637255680,    205946880,    32440320,     1966080;
  8:  277136640,  2679029760,  9267793920,  15088803840,  12882862080,   6044712960,  1557135360,   205455360,   10813440;
  9: 2217093120, 24388136960, 99766599680, 199817953280, 218680197120, 138001121280, 51154780160, 10928783360, 1239941120, 57671680;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(12*n^2+60*n+72-(8*n^2+40*n+48)*x)*P(n-2,x) + (8*n^2+16*n-24+(4*n^2+8*n-12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 24 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 24 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 + 60*n + 72 - (8*n^2 + 40*n + 48)*x)*P[n - 2, x] + (8*n^2 + 16*n - 24 + (4*n^2 + 8*n - 12)*x)*P[n - 1, x])/(n^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; v[2] = 24 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 + 60*n + 72 - (8*n^2 + 40*n + 48)*x)*v[i - 2] + (8*n^2 + 16*n - 24 + (4*n^2 + 8*n - 12)*x)*v[i - 1])/(n^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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.