Real-rooted atlas

A112493 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 - 2\,x + n\right)\,P_{n}(x) = \left(4 - 2\,x + 10\,x^{3} + 4\,x^{4} - 2\,n + 7\,n x - 7\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 2\,x^{2} - 12\,x^{3} - 10\,x^{4} - 2\,x^{5} - n x + 4\,n x^{2} + 8\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 - 3\,x + 6\,x^{2} + 2\,x^{3} + 3\,n - 7\,n x - 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 5\,x^{2} - 2\,x^{3} + n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
3510318
412617015158
5272867606301051808
6581149579610570693094525448
71214194361411184681653759009010395424784
82481438919848610540252551780289327513513501351358198688

All 15 rows

Technical details
Node id
n-d03f97987f791e57 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-acc25c6361f8bb41

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,      1;
   3:    5,     10,       3;
   4:   12,     61,      70,       15;
   5:   27,    286,     760,      630,       105;
   6:   58,   1149,    5796,    10570,      6930,       945;
   7:  121,   4194,   36141,   118468,    165375,     90090,      10395;
   8:  248,  14389,  198486,  1054025,   2551780,   2893275,    1351350,     135135;
   9:  503,  47374, 1001226,  8084450,  30618280,  58708650,   56126070,   22972950,   2027025;
  10: 1014, 151669, 4759876, 55998756, 311757160, 912241330, 1447506060, 1197296100, 436486050, 34459425;
%F Formula
The row polynomials satisfy (n-3-2*x)*P(n,x) = -(2*n-4+(2*n^2-7*n+2)*x+(n^3-6*n^2+7*n)*x^2-(3*n^2-12*n+10)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n-2)*x+(n^2-4*n+2)*x^2+(n^2-8*n+12)*x^3-(3*n-10)*x^4+2*x^5)*d/dx P(n-2,x) + (3*n-8+(2*n^2-7*n-3)*x-(5*n-6)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x+(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (2*n^2 - 7*n + 2)*x + (n^3 - 6*n^2 + 7*n)*x^2 - (3*n^2 - 12*n + 10)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n - 2)*x + (n^2 - 4*n + 2)*x^2 + (n^2 - 8*n + 12)*x^3 - (3*n - 10)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (3*n - 8 + (2*n^2 - 7*n - 3)*x - (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 - 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(2*n - 4 + (2*n^2 - 7*n + 2)*x + (n^3 - 6*n^2 + 7*n)*x^2 - (3*n^2 - 12*n + 10)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n - 2)*x + (n^2 - 4*n + 2)*x^2 + (n^2 - 8*n + 12)*x^3 - (3*n - 10)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (3*n - 8 + (2*n^2 - 7*n - 3)*x - (5*n - 6)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 - 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.