Real-rooted atlas

A008292 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-48\,x^{2} + 24\,n x + 172\,n x^{2} - 8\,n^{2} x - 160\,n^{2} x^{2} + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{2} - 72\,x^{3} - 2\,n x + 8\,n x^{2} + 232\,n x^{3} + 8\,n^{2} x^{2} - 176\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{3} - 16\,x^{4} - n x^{2} - 8\,n x^{3} + 48\,n x^{4} + 8\,n^{2} x^{3} - 32\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 + 16\,x + n - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 2\,n x + 8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3189
413233
5188144233
6120812241433
714566480435211289
81960276486348892097

All 15 rows

Technical details
Node id
n-fbeed8f9551c8453 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e202bb8437820ac7

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 1,    8;
   4: 1,   32;
   5: 1,   88,    144;
   6: 1,  208,   1224;
   7: 1,  456,   6480,    4352;
   8: 1,  960,  27648,   63488;
   9: 1, 1976, 104544,  548864,  198400;
  10: 1, 4016, 366984, 3682304, 4422400;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -((8*n^2-24*n)*x-(36*n^3-160*n^2+172*n-48)*x^2)*P(n-2,x) - (2*n*x-(8*n^2+8*n+4)*x^2-(16*n^3-176*n^2+232*n-72)*x^3)*d/dx P(n-2,x) - (n*x^2-(8*n^2-8*n+4)*x^3+(32*n^2-48*n+16)*x^4)*d^2/dx^2 P(n-2,x) + (n-1+(8*n^2-24*n+16)*x)*P(n-1,x) + ((2*n-1)*x-(8*n^2-8*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 24*n)*x - (36*n^3 - 160*n^2 + 172*n - 48)*x^2)*P[n - 2, x] - (2*n*x - (8*n^2 + 8*n + 4)*x^2 - (16*n^3 - 176*n^2 + 232*n - 72)*x^3)*D[P[n - 2, x], x] - (n*x^2 - (8*n^2 - 8*n + 4)*x^3 + (32*n^2 - 48*n + 16)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 1 + (8*n^2 - 24*n + 16)*x)*P[n - 1, x] + ((2*n - 1)*x - (8*n^2 - 8*n)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-((8*n^2 - 24*n)*x - (36*n^3 - 160*n^2 + 172*n - 48)*x^2)*v[i - 2] - (2*n*x - (8*n^2 + 8*n + 4)*x^2 - (16*n^3 - 176*n^2 + 232*n - 72)*x^3)*deriv(v[i - 2], x) - (n*x^2 - (8*n^2 - 8*n + 4)*x^3 + (32*n^2 - 48*n + 16)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 1 + (8*n^2 - 24*n + 16)*x)*v[i - 1] + ((2*n - 1)*x - (8*n^2 - 8*n)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.