Real-rooted atlas

Veronese parent of A109446 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
0112
1213
2155112
3496120
41922229164
56254030101112
611351959551131320
784912617514063141576

All 15 rows

Technical details
Node id
n-cc30e1f39dd8ee15 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1ccc0ce8bff18e11

Observed on the stored rows ?

  • Degrees: 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15 (irregular)
  • 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,  1;
  2:  1,  5,   5,   1;
  3:  4,  9,   6,   1;
  4:  1,  9,  22,  22,   9,   1;
  5:  6, 25,  40,  30,  10,   1;
  6:  1, 13,  51,  95,  95,  51,  13,   1;
  7:  8, 49, 126, 175, 140,  63,  14,   1;
  8:  1, 17,  92, 252, 406, 406, 252,  92, 17, 1;
  9: 10, 81, 288, 588, 756, 630, 336, 108, 18, 1;
%F Formula
The row polynomials satisfy (n^2+n-1+2*n*x+x^2)*P(n,x) = -(n^3-n^2-3*n+2-(n^2+5*n-1)*x-(n^3+2*n^2+3*n+2)*x^2-(2*n^2+2*n+1)*x^3-n*x^4)*P(n-1,x) + (n^2+n-1+(n^2+n-3)*x-(n^2+3*n+5)*x^2-(n^2+5*n+6)*x^3-(2*n+4)*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + x.
%t Mathematica
P[0, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 3*n + 2 - (n^2 + 5*n - 1)*x - (n^3 + 2*n^2 + 3*n + 2)*x^2 - (2*n^2 + 2*n + 1)*x^3 - n*x^4)*P[n - 1, x] + (n^2 + n - 1 + (n^2 + n - 3)*x - (n^2 + 3*n + 5)*x^2 - (n^2 + 5*n + 6)*x^3 - (2*n + 4)*x^4 - x^5)*D[P[n - 1, x], x])/(n^2 + n - 1 + 2*n*x + 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 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 3*n + 2 - (n^2 + 5*n - 1)*x - (n^3 + 2*n^2 + 3*n + 2)*x^2 - (2*n^2 + 2*n + 1)*x^3 - n*x^4)*v[i - 1] + (n^2 + n - 1 + (n^2 + n - 3)*x - (n^2 + 3*n + 5)*x^2 - (n^2 + 5*n + 6)*x^3 - (2*n + 4)*x^4 - x^5)*deriv(v[i - 1], x))/(n^2 + n - 1 + 2*n*x + 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.