Real-rooted atlas

A100861 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-8 - 48\,x - 8\,x^{2} + 4\,n + 76\,n x + 28\,n x^{2} - 38\,n^{2} x - 38\,n^{2} x^{2} + 6\,n^{3} x + 25\,n^{3} x^{2} - 8\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(4\,n - 24\,n x - 20\,n x^{2} - 10\,n^{2} x + 74\,n^{2} x^{2} + 6\,n^{3} x - 43\,n^{3} x^{2} - 3\,n^{3} x^{3} + 4\,n^{4} x^{2} + 7\,n^{4} x^{3} + n^{5} x^{2} - 5\,n^{5} x^{3} + n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-8\,n x^{2} - 8\,n x^{3} + 28\,n^{2} x^{3} - 12\,n^{3} x^{3} - 2\,n^{3} x^{4} + 4\,n^{4} x^{4} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
333
462430
510120130
615360405780
72184028353696
828168011340672019768
9363024340206048097560
1045504085050302400118125510660
11557920187110110880012993752603260
12661188037422033264007796250224532013754136
1378171606949808648640337837502918916072333768
14912402412162152018016011824312520432412046351305390339040
Technical details
Node id
n-6672882b678a38a4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-00a4659bce2e29b7

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:  0;
  2:  1;
  3:  3;
  4:  6,   24;
  5: 10,  120;
  6: 15,  360,   405;
  7: 21,  840,  2835;
  8: 28, 1680, 11340,  6720;
  9: 36, 3024, 34020, 60480;
%F Formula
The row polynomials satisfy (4*n-8+(6*n^3-38*n^2+76*n-48)*x+(n^5-8*n^4+25*n^3-38*n^2+28*n-8)*x^2)*P(n,x) = (4*n+(6*n^3-10*n^2-24*n)*x+(n^5+4*n^4-43*n^3+74*n^2-20*n)*x^2+(n^6-5*n^5+7*n^4-3*n^3)*x^3)*P(n-1,x) - (8*n*x^2+(12*n^3-28*n^2+8*n)*x^3+(2*n^5-4*n^4+2*n^3)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + (6*n^3 - 10*n^2 - 24*n)*x + (n^5 + 4*n^4 - 43*n^3 + 74*n^2 - 20*n)*x^2 + (n^6 - 5*n^5 + 7*n^4 - 3*n^3)*x^3)*P[n - 1, x] - (8*n*x^2 + (12*n^3 - 28*n^2 + 8*n)*x^3 + (2*n^5 - 4*n^4 + 2*n^3)*x^4)*D[P[n - 1, x], x])/(4*n - 8 + (6*n^3 - 38*n^2 + 76*n - 48)*x + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*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] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((4*n + (6*n^3 - 10*n^2 - 24*n)*x + (n^5 + 4*n^4 - 43*n^3 + 74*n^2 - 20*n)*x^2 + (n^6 - 5*n^5 + 7*n^4 - 3*n^3)*x^3)*v[i - 1] - (8*n*x^2 + (12*n^3 - 28*n^2 + 8*n)*x^3 + (2*n^5 - 4*n^4 + 2*n^3)*x^4)*deriv(v[i - 1], x))/(4*n - 8 + (6*n^3 - 38*n^2 + 76*n - 48)*x + (n^5 - 8*n^4 + 25*n^3 - 38*n^2 + 28*n - 8)*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

Same fingerprint ?

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.