Real-rooted atlas

A115951 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 4\,n x + 12\,n^{2} + 6\,n^{2} x + 13\,n^{3} + 6\,n^{3} x + 4\,n^{3} x^{2} + 6\,n^{4} + n^{5}\right)\,P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3} + 16\,n + 100\,n x + 36\,n x^{2} + 16\,n x^{3} + 2\,n^{2} + 106\,n^{2} x + 12\,n^{2} x^{2} + 8\,n^{2} x^{3} - 14\,n^{3} + 48\,n^{3} x - 8\,n^{3} x^{2} - 10\,n^{4} + 10\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(16\,x + 48\,x^{2} + 48\,x^{3} + 16\,x^{4} + 48\,n x + 104\,n x^{2} + 88\,n x^{3} + 32\,n x^{4} + 52\,n^{2} x + 76\,n^{2} x^{2} + 40\,n^{2} x^{3} + 16\,n^{2} x^{4} + 24\,n^{3} x + 24\,n^{3} x^{2} + 4\,n^{4} x + 4\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1088
2085462
300108320428
4005496017502764
50009607000907217032
6000320105004536045276101456
70000700090720271656219648589024

All 15 rows

Technical details
Node id
n-8628db06a7bc9bde · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7b50549dba03cef1

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: 0, 8;
  2: 0, 8,  54;
  3: 0, 0, 108, 320;
  4: 0, 0,  54, 960,  1750;
  5: 0, 0,   0, 960,  7000,  9072;
  6: 0, 0,   0, 320, 10500, 45360,  45276;
  7: 0, 0,   0,   0,  7000, 90720, 271656,  219648;
  8: 0, 0,   0,   0,  1750, 90720, 679140, 1537536, 1042470;
  9: 0, 0,   0,   0,     0, 45360, 905520, 4612608, 8339760, 4862000;
%F Formula
The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n+(6*n^3+6*n^2-4*n)*x+4*n^3*x^2)*P(n,x) = -(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8-(10*n^4+48*n^3+106*n^2+100*n+24)*x+(8*n^3-12*n^2-36*n-24)*x^2-(8*n^2+16*n+8)*x^3)*P(n-1,x) + ((4*n^4+24*n^3+52*n^2+48*n+16)*x+(4*n^4+24*n^3+76*n^2+104*n+48)*x^2+(40*n^2+88*n+48)*x^3+(16*n^2+32*n+16)*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[(-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (10*n^4 + 48*n^3 + 106*n^2 + 100*n + 24)*x + (8*n^3 - 12*n^2 - 36*n - 24)*x^2 - (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (4*n^4 + 24*n^3 + 76*n^2 + 104*n + 48)*x^2 + (40*n^2 + 88*n + 48)*x^3 + (16*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n + (6*n^3 + 6*n^2 - 4*n)*x + 4*n^3*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] = (-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (10*n^4 + 48*n^3 + 106*n^2 + 100*n + 24)*x + (8*n^3 - 12*n^2 - 36*n - 24)*x^2 - (8*n^2 + 16*n + 8)*x^3)*v[i - 1] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (4*n^4 + 24*n^3 + 76*n^2 + 104*n + 48)*x^2 + (40*n^2 + 88*n + 48)*x^3 + (16*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n + (6*n^3 + 6*n^2 - 4*n)*x + 4*n^3*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.