Real-rooted atlas

A113413 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(1 + 4\,x^{2} + 16\,x^{3} + n + 3\,n x - 12\,n x^{2} - 24\,n x^{3} + 3\,n^{2} x + 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-x + 5\,x^{2} + 36\,x^{3} + 32\,x^{4} - n x - 10\,n x^{2} - 24\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 9\,x^{3} + 16\,x^{4} + 8\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
216714
319231750
4112487641178
51158220423399634
61181254287656822392258
71211777751907264919355778042

All 15 rows

Technical details
Node id
n-f239188c3594f81d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b42d277a4b716fed

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,  3;
  2: 1,  6,   7;
  3: 1,  9,  23,   17;
  4: 1, 12,  48,   76,   41;
  5: 1, 15,  82,  204,  233,    99;
  6: 1, 18, 125,  428,  765,   682,   239;
  7: 1, 21, 177,  775, 1907,  2649,  1935,   577;
  8: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;
  9: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n+1+(3*n^2+3*n)*x+(8*n^2-12*n+4)*x^2+(8*n^2-24*n+16)*x^3)*P(n-1,x) - ((n+1)*x+(10*n-5)*x^2+(24*n-36)*x^3+(16*n-32)*x^4)*d/dx P(n-1,x) + (x^2+9*x^3+16*x^4+8*x^5)*d^2/dx^2 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 + 1 + (3*n^2 + 3*n)*x + (8*n^2 - 12*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - ((n + 1)*x + (10*n - 5)*x^2 + (24*n - 36)*x^3 + (16*n - 32)*x^4)*D[P[n - 1, x], x] + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*D[P[n - 1, x], {x, 2}])/(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; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (3*n^2 + 3*n)*x + (8*n^2 - 12*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3)*v[i - 1] - ((n + 1)*x + (10*n - 5)*x^2 + (24*n - 36)*x^3 + (16*n - 32)*x^4)*deriv(v[i - 1], x) + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*deriv(deriv(v[i - 1], x), x))/(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

Candidate equalities ?

Same fingerprint ?

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