Real-rooted atlas

A081579 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 8\,n x + n^{2} + 12\,n^{3} x + 16\,n^{3} x^{2} + 16\,n^{4} x^{2}\right)\,P_{n}(x) = \left(-32\,x - 64\,x^{2} - 512\,x^{3} + n - 8\,n x - 224\,n x^{2} - 512\,n x^{3} + n^{2} + 32\,n^{2} x + 128\,n^{2} x^{2} + 384\,n^{2} x^{3} + 12\,n^{3} x + 176\,n^{3} x^{2} + 512\,n^{3} x^{3} + 16\,n^{4} x^{2} + 128\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-x - 24\,x^{2} - 192\,x^{3} - 1024\,x^{4} - n x - 40\,n x^{2} - 512\,n x^{3} - 2048\,n x^{4} - 12\,n^{2} x^{2} - 272\,n^{2} x^{3} - 1280\,n^{2} x^{4} - 16\,n^{3} x^{3} - 256\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
211617
313233
4148144193
5164432497
618086410241969
7196144040965633
81112216010240640018913
911283024204803200055633
1011444032358409600036864172881
111160518457344224000221184507873
1211766480860164480007741442007041515521
1311927920122880806400206438414049284406705
1412089504168960134400046448645619712104857612835825
Technical details
Node id
n-4a49127fe76d153d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3826063dc2ff7965

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: 1,  16;
  3: 1,  32;
  4: 1,  48,  144;
  5: 1,  64,  432;
  6: 1,  80,  864,  1024;
  7: 1,  96, 1440,  4096;
  8: 1, 112, 2160, 10240,  6400;
  9: 1, 128, 3024, 20480, 32000;
%F Formula
The row polynomials satisfy (n^2+n+(12*n^3-8*n)*x+(16*n^4+16*n^3)*x^2)*P(n,x) = (n^2+n+(12*n^3+32*n^2-8*n-32)*x+(16*n^4+176*n^3+128*n^2-224*n-64)*x^2+(128*n^4+512*n^3+384*n^2-512*n-512)*x^3)*P(n-1,x) - ((n+1)*x+(12*n^2+40*n+24)*x^2+(16*n^3+272*n^2+512*n+192)*x^3+(256*n^3+1280*n^2+2048*n+1024)*x^4)*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[((n^2 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*P[n - 1, x] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*x^2)]];
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] = ((n^2 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*v[i - 1] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*x^2)); 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. [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.