Real-rooted atlas

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

nonnegative coefficients

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

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(32\,x - 2\,n^{2} - 10\,n^{2} x^{2} - 3\,n^{3} + 4\,n^{3} x - 13\,n^{3} x^{2} - n^{4} - 4\,n^{4} x + 9\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(8\,x + 40\,x^{2} + 2\,n^{2} x - 16\,n^{2} x^{2} - 18\,n^{2} x^{3} + 4\,n^{3} x + 4\,n^{3} x^{2} - 32\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(8\,x^{2} + 8\,x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3} - 4\,n^{2} x^{4} - 8\,n^{3} x^{3} - 8\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(2\,n + 4\,n^{2} - 4\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x - 4\,n^{2} x - 4\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
26814
3246488
4120480144744
5720384024487008
650403360033264435276256
740320322560435456126976925312
836288033868805806080270336019840012457600
936288003870720080559360521011208844800183841280
10399168004790016001173553920976588800270758400127365122952556032
1147900160063866880001801046016018377441280718671360080525721651245561856
12622702080091329638400291424573440352772259840179507328000334623744001096044544955819239424
1387178291200139485265920049691625984006966883123200438428390400011678117068809328410828819063456391168
1413076743680002266635571200089183391897600142208549683200106890271488000374059026432005048416794624121840467968404832403054592
Technical details
Node id
n-9c1df38ba8682fc5 · 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 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:       2;
  2:       6,        8;
  3:      24,       64;
  4:     120,      480,      144;
  5:     720,     3840,     2448;
  6:    5040,    33600,    33264,     4352;
  7:   40320,   322560,   435456,   126976;
  8:  362880,  3386880,  5806080,  2703360,  198400;
  9: 3628800, 38707200, 80559360, 52101120, 8844800;
%F Formula
The row polynomials satisfy n^2*P(n,x) = -(n^4+3*n^3+2*n^2+(4*n^4-4*n^3-32)*x-(9*n^4-13*n^3-10*n^2)*x^2)*P(n-2,x) + ((4*n^3+2*n^2+8)*x+(4*n^4+4*n^3-16*n^2+40)*x^2+(4*n^4-32*n^3-18*n^2)*x^3)*d/dx P(n-2,x) - ((4*n^2-8)*x^2+(8*n^3+8*n^2-8)*x^3+(8*n^3+4*n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3+4*n^2+2*n+(4*n^3-4*n^2)*x)*P(n-1,x) - ((4*n^2-4)*x+(4*n^3+4*n^2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 3*n^3 + 2*n^2 + (4*n^4 - 4*n^3 - 32)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*P[n - 2, x] + ((4*n^3 + 2*n^2 + 8)*x + (4*n^4 + 4*n^3 - 16*n^2 + 40)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 8)*x^2 + (8*n^3 + 8*n^2 - 8)*x^3 + (8*n^3 + 4*n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 + 4*n^2 + 2*n + (4*n^3 - 4*n^2)*x)*P[n - 1, x] - ((4*n^2 - 4)*x + (4*n^3 + 4*n^2)*x^2)*D[P[n - 1, x], x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 3*n^3 + 2*n^2 + (4*n^4 - 4*n^3 - 32)*x - (9*n^4 - 13*n^3 - 10*n^2)*x^2)*v[i - 2] + ((4*n^3 + 2*n^2 + 8)*x + (4*n^4 + 4*n^3 - 16*n^2 + 40)*x^2 + (4*n^4 - 32*n^3 - 18*n^2)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 8)*x^2 + (8*n^3 + 8*n^2 - 8)*x^3 + (8*n^3 + 4*n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 + 4*n^2 + 2*n + (4*n^3 - 4*n^2)*x)*v[i - 1] - ((4*n^2 - 4)*x + (4*n^3 + 4*n^2)*x^2)*deriv(v[i - 1], x))/(n^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.