Real-rooted atlas

A124324 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(12\,x^{2} - 16\,n x^{2} + 7\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-4}(x) + \left(-8\,x + 6\,x^{2} + 8\,n x - 5\,n x^{2} - 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-4\,x^{2} + 4\,n x^{2} - n^{2} x^{2}\right)\,P_{n-3}'(x) + \left(4 + 4\,x - 2\,n - 7\,n x + 2\,n^{2} x\right)\,P_{n-2}(x) + \left(2\,x - 3\,x^{2} - n x + n x^{2}\right)\,P_{n-2}'(x) + \left(-8 + 3\,x + 3\,n - n x\right)\,P_{n-1}(x) + \left(-3\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)
\(P_{3}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2213
3549
41217332
5277625128
65831717515565
7121121211762102719
824843217287217010514131
9503146444117520188220578715
1010144788521485117090531500945467100
11203715269210531181316150386155277202937872
1240844783374921675932003642438555232151039519501597
13817914798042218581361587240422879608168160405405136122561
14163704537405973084713850676833864941081126024909774765135135995936427
Technical details
Node id
n-6e2c2e6550c373d3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b31d2a49add81a29

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:   2,     1;
  3:   5,     4;
  4:  12,    17,     3;
  5:  27,    76,    25;
  6:  58,   317,   175,    15;
  7: 121,  1212,  1176,   210;
  8: 248,  4321,  7287,  2170,  105;
  9: 503, 14644, 41175, 20188, 2205;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(n^3-7*n^2+16*n-12)*x^2*P(n-4,x) - ((2*n^2-8*n+8)*x-(n^2-5*n+6)*x^2)*P(n-3,x) - (n^2-4*n+4)*x^2*d/dx P(n-3,x) - (2*n-4-(2*n^2-7*n+4)*x)*P(n-2,x) - ((n-2)*x-(n-3)*x^2)*d/dx P(n-2,x) + (3*n-8-(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[3, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 16*n - 12)*x^2*P[n - 4, x] - ((2*n^2 - 8*n + 8)*x - (n^2 - 5*n + 6)*x^2)*P[n - 3, x] - (n^2 - 4*n + 4)*x^2*D[P[n - 3, x], x] - (2*n - 4 - (2*n^2 - 7*n + 4)*x)*P[n - 2, x] - ((n - 2)*x - (n - 3)*x^2)*D[P[n - 2, x], x] + (3*n - 8 - (n - 3)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];
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; v[3] = 2 + x; v[4] = 5 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - 7*n^2 + 16*n - 12)*x^2*v[i - 4] - ((2*n^2 - 8*n + 8)*x - (n^2 - 5*n + 6)*x^2)*v[i - 3] - (n^2 - 4*n + 4)*x^2*deriv(v[i - 3], x) - (2*n - 4 - (2*n^2 - 7*n + 4)*x)*v[i - 2] - ((n - 2)*x - (n - 3)*x^2)*deriv(v[i - 2], x) + (3*n - 8 - (n - 3)*x)*v[i - 1] + (n - 3)*x*deriv(v[i - 1], x))/(n - 3)); 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.