Real-rooted atlas

A113413 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(4 + 10\,x + 8\,x^{2} - 6\,n - 8\,n x + 2\,n^{2} - 4\,n^{2} x - 8\,n^{2} x^{2} + 2\,n^{3} x\right)\,P_{n}(x) = \left(8 + 32\,x + 16\,x^{2} - 28\,n - 150\,n x - 216\,n x^{2} - 96\,n x^{3} + 16\,n^{2} + 70\,n^{2} x + 56\,n^{2} x^{2} - 2\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(1 + 8\,x + 18\,x^{2} + 8\,x^{3} - 3\,n - 28\,n x - 86\,n x^{2} - 96\,n x^{3} - 32\,n x^{4} + n^{2} + 8\,n^{2} x + 18\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2243256
392288192572
43041536230410245168
593265281574415360512043684
6272824480821761280009216024576354120
77740847683661447936009062405160961146882789276

All 15 rows

Technical details
Node id
n-306ebc95b6773536 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-766f0e1c214c9991

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     4;
  2:    24,     32;
  3:    92,    288,     192;
  4:   304,   1536,    2304,     1024;
  5:   932,   6528,   15744,    15360,     5120;
  6:  2728,  24480,   82176,   128000,    92160,    24576;
  7:  7740,  84768,  366144,   793600,   906240,   516096,   114688;
  8: 21472, 277760, 1469952,  4106240,  6512640,  5849088,  2752512,   524288;
  9: 58564, 873984, 5481216, 18798592, 38430720, 47824896, 35323904, 14155776, 2359296;
%F Formula
The row polynomials satisfy (2*n^2-6*n+4+(2*n^3-4*n^2-8*n+10)*x-(8*n^2-8)*x^2)*P(n,x) = -(2*n^3-16*n^2+28*n-8+(6*n^3-70*n^2+150*n-32)*x-(56*n^2-216*n+16)*x^2+96*n*x^3)*P(n-1,x) + (n^2-3*n+1+(8*n^2-28*n+8)*x+(18*n^2-86*n+18)*x^2+(8*n^2-96*n+8)*x^3-32*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 16*n^2 + 28*n - 8 + (6*n^3 - 70*n^2 + 150*n - 32)*x - (56*n^2 - 216*n + 16)*x^2 + 96*n*x^3)*P[n - 1, x] + (n^2 - 3*n + 1 + (8*n^2 - 28*n + 8)*x + (18*n^2 - 86*n + 18)*x^2 + (8*n^2 - 96*n + 8)*x^3 - 32*n*x^4)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 + (2*n^3 - 4*n^2 - 8*n + 10)*x - (8*n^2 - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 16*n^2 + 28*n - 8 + (6*n^3 - 70*n^2 + 150*n - 32)*x - (56*n^2 - 216*n + 16)*x^2 + 96*n*x^3)*v[i - 1] + (n^2 - 3*n + 1 + (8*n^2 - 28*n + 8)*x + (18*n^2 - 86*n + 18)*x^2 + (8*n^2 - 96*n + 8)*x^3 - 32*n*x^4)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 + (2*n^3 - 4*n^2 - 8*n + 10)*x - (8*n^2 - 8)*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.