Real-rooted atlas

A008292 → γ (scale 1/4) → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(24\,x + 12\,x^{2} - n + 28\,n x - 55\,n x^{2} + n^{2} - 36\,n^{2} x + 83\,n^{2} x^{2} + n^{3} + 20\,n^{3} x - 49\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x + 24\,x^{2} + 18\,x^{3} + 8\,n x + 28\,n x^{2} - 76\,n x^{3} - 10\,n^{2} x - 4\,n^{2} x^{2} + 102\,n^{2} x^{3} + 4\,n^{3} x - 12\,n^{3} x^{2} - 48\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} + 8\,x^{3} + 4\,x^{4} + 8\,n x^{2} - 8\,n x^{3} - 16\,n x^{4} - 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 20\,n^{2} x^{4} - 8\,n^{3} x^{3} - 8\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(-8\,x + 20\,n x - 2\,n^{2} - 16\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,n x - 4\,n x^{2} - 4\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
111
222
36814
4246488
5120480144744
6720384024487008
750403360033264435276256
840320322560435456126976925312

All 15 rows

Technical details
Node id
n-0f03351624a7d9e5 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4207ccd284934a16

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   3:       6,        8;
   4:      24,       64;
   5:     120,      480,      144;
   6:     720,     3840,     2448;
   7:    5040,    33600,    33264,     4352;
   8:   40320,   322560,   435456,   126976;
   9:  362880,  3386880,  5806080,  2703360,  198400;
  10: 3628800, 38707200, 80559360, 52101120, 8844800;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^4-n^3-n^2+n+(4*n^4-20*n^3+36*n^2-28*n-24)*x-(9*n^4-49*n^3+83*n^2-55*n+12)*x^2)*P(n-2,x) + ((4*n^3-10*n^2+8*n+6)*x+(4*n^4-12*n^3-4*n^2+28*n+24)*x^2+(4*n^4-48*n^3+102*n^2-76*n+18)*x^3)*d/dx P(n-2,x) - ((4*n^2-8*n-4)*x^2+(8*n^3-16*n^2+8*n-8)*x^3+(8*n^3-20*n^2+16*n-4)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-2*n^2+(4*n^3-16*n^2+20*n-8)*x)*P(n-1,x) - ((4*n^2-8*n)*x+(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - n^3 - n^2 + n + (4*n^4 - 20*n^3 + 36*n^2 - 28*n - 24)*x - (9*n^4 - 49*n^3 + 83*n^2 - 55*n + 12)*x^2)*P[n - 2, x] + ((4*n^3 - 10*n^2 + 8*n + 6)*x + (4*n^4 - 12*n^3 - 4*n^2 + 28*n + 24)*x^2 + (4*n^4 - 48*n^3 + 102*n^2 - 76*n + 18)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 8*n - 4)*x^2 + (8*n^3 - 16*n^2 + 8*n - 8)*x^3 + (8*n^3 - 20*n^2 + 16*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 2*n^2 + (4*n^3 - 16*n^2 + 20*n - 8)*x)*P[n - 1, x] - ((4*n^2 - 8*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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; for(n = 3, N, my(i = n); v[i] = (-(n^4 - n^3 - n^2 + n + (4*n^4 - 20*n^3 + 36*n^2 - 28*n - 24)*x - (9*n^4 - 49*n^3 + 83*n^2 - 55*n + 12)*x^2)*v[i - 2] + ((4*n^3 - 10*n^2 + 8*n + 6)*x + (4*n^4 - 12*n^3 - 4*n^2 + 28*n + 24)*x^2 + (4*n^4 - 48*n^3 + 102*n^2 - 76*n + 18)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 8*n - 4)*x^2 + (8*n^3 - 16*n^2 + 8*n - 8)*x^3 + (8*n^3 - 20*n^2 + 16*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 2*n^2 + (4*n^3 - 16*n^2 + 20*n - 8)*x)*v[i - 1] - ((4*n^2 - 8*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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.