Real-rooted atlas

A145596 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-16 + 16\,x - 4\,x^{2} + 20\,n - 20\,n x + 3\,n x^{2} - 6\,n^{2} + 2\,n^{2} x + n^{2} x^{2} + n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(24 - 32\,x + 14\,x^{2} - 2\,x^{3} + 18\,n + 4\,n x - 10\,n x^{2} + 4\,n x^{3} + 11\,n^{2} + 8\,n^{2} x + 3\,n^{2} x^{3} + 6\,n^{3} + 2\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(7\,x - 13\,x^{2} + 9\,x^{3} - 3\,x^{4} + 4\,n x - 2\,n x^{2} - 2\,n x^{4} + 2\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
214620
3606412136
419039018020780
549016801470400304070
610925684823242007704220020
721841612835280295681008013445694640

All 15 rows

Technical details
Node id
n-c3dbf3208ee5665d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a5d67d157cd7d6b6

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:    2;
  2:   14,     6;
  3:   60,    64,     12;
  4:  190,   390,    180,     20;
  5:  490,  1680,   1470,    400,     30;
  6: 1092,  5684,   8232,   4200,    770,     42;
  7: 2184, 16128,  35280,  29568,  10080,   1344,    56;
  8: 4020, 40068, 123984, 156240,  86940,  21420,  2184,   72;
  9: 6930, 89760, 374220, 665280, 554400, 221760, 41580, 3360, 90;
%F Formula
The row polynomials satisfy (n^4+n^3-6*n^2+20*n-16+(2*n^3+2*n^2-20*n+16)*x+(n^2+3*n-4)*x^2)*P(n,x) = (n^4+6*n^3+11*n^2+18*n+24+(3*n^4+2*n^3+8*n^2+4*n-32)*x+(6*n^3-10*n+14)*x^2+(3*n^2+4*n-2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2+4*n+7)*x-(2*n^3-2*n^2+2*n+13)*x^2-(4*n^2-9)*x^3-(2*n+3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^4 + 6*n^3 + 11*n^2 + 18*n + 24 + (3*n^4 + 2*n^3 + 8*n^2 + 4*n - 32)*x + (6*n^3 - 10*n + 14)*x^2 + (3*n^2 + 4*n - 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 + 4*n + 7)*x - (2*n^3 - 2*n^2 + 2*n + 13)*x^2 - (4*n^2 - 9)*x^3 - (2*n + 3)*x^4)*D[P[n - 1, x], x])/(n^4 + n^3 - 6*n^2 + 20*n - 16 + (2*n^3 + 2*n^2 - 20*n + 16)*x + (n^2 + 3*n - 4)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 6*n^3 + 11*n^2 + 18*n + 24 + (3*n^4 + 2*n^3 + 8*n^2 + 4*n - 32)*x + (6*n^3 - 10*n + 14)*x^2 + (3*n^2 + 4*n - 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2 + 4*n + 7)*x - (2*n^3 - 2*n^2 + 2*n + 13)*x^2 - (4*n^2 - 9)*x^3 - (2*n + 3)*x^4)*deriv(v[i - 1], x))/(n^4 + n^3 - 6*n^2 + 20*n - 16 + (2*n^3 + 2*n^2 - 20*n + 16)*x + (n^2 + 3*n - 4)*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.