Real-rooted atlas

A145596 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x^{2} + 9\,n^{2} + 8\,n^{2} x + n^{2} x^{2} + 6\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(12 + 6\,x + 8\,x^{2} + 2\,x^{3} + 38\,n + 29\,n x + 19\,n x^{2} + 5\,n x^{3} + 44\,n^{2} + 41\,n^{2} x + 17\,n^{2} x^{2} + 3\,n^{2} x^{3} + 22\,n^{3} + 21\,n^{3} x + 6\,n^{3} x^{2} + 4\,n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(-6\,x - 10\,x^{2} - 6\,x^{3} - 2\,x^{4} - 13\,n x - 21\,n x^{2} - 10\,n x^{3} - 2\,n x^{4} - 9\,n^{2} x - 13\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
22820351
3120136444304
44957804058051765
5200240702970950130610128
68008200201901986241925196757799
73182494640111384662482111235282808329024

All 15 rows

Technical details
Node id
n-2d75fda34284dcb7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e6f22d789fa4c102

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:      6,       2;
  2:     28,      20,       3;
  3:    120,     136,      44,       4;
  4:    495,     780,     405,      80,       5;
  5:   2002,    4070,    2970,     950,     130,      6;
  6:   8008,   20020,   19019,    8624,    1925,    196,     7;
  7:  31824,   94640,  111384,   66248,   21112,   3528,   280,    8;
  8: 125970,  434928,  612612,  454272,  191100,  45864,  6006,  384,   9;
  9: 497420, 1957380, 3217080, 2870280, 1513680, 482160, 91140, 9660, 510, 10;
%F Formula
The row polynomials satisfy (n^4+6*n^3+9*n^2+4*n+(2*n^3+8*n^2)*x+(n^2+4*n)*x^2)*P(n,x) = (4*n^4+22*n^3+44*n^2+38*n+12+(3*n^4+21*n^3+41*n^2+29*n+6)*x+(6*n^3+17*n^2+19*n+8)*x^2+(3*n^2+5*n+2)*x^3)*P(n-1,x) - ((2*n^3+9*n^2+13*n+6)*x+(2*n^3+13*n^2+21*n+10)*x^2+(4*n^2+10*n+6)*x^3+(2*n+2)*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[((4*n^4 + 22*n^3 + 44*n^2 + 38*n + 12 + (3*n^4 + 21*n^3 + 41*n^2 + 29*n + 6)*x + (6*n^3 + 17*n^2 + 19*n + 8)*x^2 + (3*n^2 + 5*n + 2)*x^3)*P[n - 1, x] - ((2*n^3 + 9*n^2 + 13*n + 6)*x + (2*n^3 + 13*n^2 + 21*n + 10)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 6*n^3 + 9*n^2 + 4*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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] = ((4*n^4 + 22*n^3 + 44*n^2 + 38*n + 12 + (3*n^4 + 21*n^3 + 41*n^2 + 29*n + 6)*x + (6*n^3 + 17*n^2 + 19*n + 8)*x^2 + (3*n^2 + 5*n + 2)*x^3)*v[i - 1] - ((2*n^3 + 9*n^2 + 13*n + 6)*x + (2*n^3 + 13*n^2 + 21*n + 10)*x^2 + (4*n^2 + 10*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 6*n^3 + 9*n^2 + 4*n + (2*n^3 + 8*n^2)*x + (n^2 + 4*n)*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.