Real-rooted atlas

A088996 → basis A013610 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 18\,n x - 9\,n x^{2} + 12\,n^{2} + 30\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{3} - 12\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 6\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
23305487
3152046756481542
41051788839715336972035346
594519242115803305964375030174960991944
610395245682179336761448761100952010025208367416032903208
7135135363045631052754130097664304426755407767608293381676881798401258671888

All 15 rows

Technical details
Node id
n-c0d78f14ade63b14 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8d371bb5e40cee22

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:        1,          6;
  2:        3,         30,          54;
  3:       15,        204,         675,         648;
  4:      105,       1788,        8397,       15336,         9720;
  5:      945,      19242,      115803,      305964,       375030,       174960;
  6:    10395,     245682,     1793367,     6144876,     11009520,     10025208,      3674160;
  7:   135135,    3630456,    31052754,   130097664,    304426755,    407767608,    293381676,     88179840;
  8:  2027025,   60943320,   596487726,  2945153088,   8469331245,  14884004664,  15818218164,   9369737856,   2380855680;
  9: 34459425, 1145495790, 12608495802, 71541160632, 243377547885, 527434411626, 737755855704, 647390561184, 325080333936, 71425670400;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(12*n^3-30*n^2+18*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (12*n^3 - 30*n^2 + 18*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(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] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (12*n^3 - 30*n^2 + 18*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.