Real-rooted atlas

A178821 → 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

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-144 + 96\,x - 84\,n + 56\,n x - 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-2}(x) + \left(-32 - 16\,x + 24\,n + 12\,n x + 8\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 30 + 20\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1302050
25409602401740
375602296014560224047320
490720394240439040161280179201103200
5979776555609691929605806080148377612902423147712
69797760685977601532697601434316806171648012042240860160449715840
7923788807698662402185497600277668864017524531205663539208921088054067208237856000

All 15 rows

Technical details
Node id
n-9a0b62b1efd950a7 · 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:         30,          20;
  2:        540,         960,          240;
  3:       7560,       22960,        14560,         2240;
  4:      90720,      394240,       439040,       161280,        17920;
  5:     979776,     5556096,      9192960,      5806080,      1483776,       129024;
  6:    9797760,    68597760,    153269760,    143431680,     61716480,     12042240,       860160;
  7:   92378880,   769866240,   2185497600,   2776688640,   1752453120,    566353920,     89210880,     5406720;
  8:  831409920,  8037089280,  27803381760,  45266411520,  38648586240,  18134138880,   4671406080,   616366080,   32440320;
  9: 7205552640, 79261445120, 324241448960, 649408348160, 710710640640, 448503644160, 166253035520, 35518545920, 4029808640, 187432960;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(12*n^2+84*n+144-(8*n^2+56*n+96)*x)*P(n-2,x) + (8*n^2+24*n-32+(4*n^2+12*n-16)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 30 + 20*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 30 + 20*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 + 84*n + 144 - (8*n^2 + 56*n + 96)*x)*P[n - 2, x] + (8*n^2 + 24*n - 32 + (4*n^2 + 12*n - 16)*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] = 30 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 + 84*n + 144 - (8*n^2 + 56*n + 96)*x)*v[i - 2] + (8*n^2 + 24*n - 32 + (4*n^2 + 12*n - 16)*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.