Real-rooted atlas

A062145 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 - 8\,x - 4\,n - 12\,n x + 3\,n^{2} + 6\,n^{2} x + 2\,n^{3} + 11\,n^{3} x - n^{4} - 3\,n^{4} x - 3\,n^{5} x + n^{6} x\right)\,P_{n}(x) = \left(-4 + 8\,x + 32\,x^{2} - 32\,n x - 58\,n x^{2} + 5\,n^{2} + 28\,n^{2} x + 17\,n^{2} x^{2} + 9\,n^{3} x + 23\,n^{3} x^{2} - n^{4} - 13\,n^{4} x - 16\,n^{4} x^{2} - n^{5} x - n^{5} x^{2} + n^{6} x + 3\,n^{6} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 8\,x^{2} + 10\,x^{3} + 5\,n x + 12\,n x^{2} - 17\,n x^{3} - n^{2} x + n^{2} x^{3} - 2\,n^{3} x - 8\,n^{3} x^{2} + 10\,n^{3} x^{3} + 2\,n^{4} x^{2} - 2\,n^{4} x^{3} + 2\,n^{5} x^{2} - 2\,n^{5} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3108090
418360540918
5281008378022407056
64022401512022400700046800
7544320453601209609450018144283338
8707560113400470400661500317520411601611610

All 15 rows

Technical details
Node id
n-cd3bd46c9605084a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fb006a6f3bd92843

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   0;
   2:   4;
   3:  10,    80;
   4:  18,   360,    540;
   5:  28,  1008,   3780,    2240;
   6:  40,  2240,  15120,   22400,     7000;
   7:  54,  4320,  45360,  120960,    94500,    18144;
   8:  70,  7560, 113400,  470400,   661500,   317520,   41160;
   9:  88, 12320, 249480, 1478400,  3234000,  2794176,  905520,   84480;
  10: 108, 19008, 498960, 3991680, 12474000, 16765056, 9779616, 2280960, 160380;
%F Formula
The row polynomials satisfy -(n^4-2*n^3-3*n^2+4*n+4-(n^6-3*n^5-3*n^4+11*n^3+6*n^2-12*n-8)*x)*P(n,x) = -(n^4-5*n^2+4-(n^6-n^5-13*n^4+9*n^3+28*n^2-32*n+8)*x-(3*n^6-n^5-16*n^4+23*n^3+17*n^2-58*n+32)*x^2)*P(n-1,x) - ((2*n^3+n^2-5*n+2)*x-(2*n^5+2*n^4-8*n^3+12*n-8)*x^2+(2*n^5+2*n^4-10*n^3-n^2+17*n-10)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 5*n^2 + 4 - (n^6 - n^5 - 13*n^4 + 9*n^3 + 28*n^2 - 32*n + 8)*x - (3*n^6 - n^5 - 16*n^4 + 23*n^3 + 17*n^2 - 58*n + 32)*x^2)*P[n - 1, x] - ((2*n^3 + n^2 - 5*n + 2)*x - (2*n^5 + 2*n^4 - 8*n^3 + 12*n - 8)*x^2 + (2*n^5 + 2*n^4 - 10*n^3 - n^2 + 17*n - 10)*x^3)*D[P[n - 1, x], x])/(-n^4 + 2*n^3 + 3*n^2 - 4*n - 4 + (n^6 - 3*n^5 - 3*n^4 + 11*n^3 + 6*n^2 - 12*n - 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = (-(n^4 - 5*n^2 + 4 - (n^6 - n^5 - 13*n^4 + 9*n^3 + 28*n^2 - 32*n + 8)*x - (3*n^6 - n^5 - 16*n^4 + 23*n^3 + 17*n^2 - 58*n + 32)*x^2)*v[i - 1] - ((2*n^3 + n^2 - 5*n + 2)*x - (2*n^5 + 2*n^4 - 8*n^3 + 12*n - 8)*x^2 + (2*n^5 + 2*n^4 - 10*n^3 - n^2 + 17*n - 10)*x^3)*deriv(v[i - 1], x))/(-n^4 + 2*n^3 + 3*n^2 - 4*n - 4 + (n^6 - 3*n^5 - 3*n^4 + 11*n^3 + 6*n^2 - 12*n - 8)*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

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.