Real-rooted atlas

A249247 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-4}(x) + \left(4\,x^{3}\right)\,P_{n-4}'(x) + \left(8\,x - 4\,n x\right)\,P_{n-3}(x) + \left(-8\,x^{2} + 4\,n x^{2}\right)\,P_{n-3}'(x) + \left(-2 + 4\,x + 3\,n - n^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2}\right)\,P_{n-2}'(x) + \left(-2 + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 2 + 12\,x\)
\(P_{3}(x) = 6 + 40\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
221214
3640955
42416863255
5120864405161405
6720528028082089016
75040374402160021762566281
84032030240018468022400525550325
9362880274176017463602400008125365099161
10362880027578880181440002728320116250111652197366
11399168003048192002057529603311616016537502397649585282895
12479001600367234560025310880004296499202410800045511221077136650339
136227020800479001600003358998720059512320003651480008297856600256494041905945
1487178291200672518246400478522598400878169600005783400000150885504145980836481331971844960
Technical details
Node id
n-eb1635a17d0cadb8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-84c4a564182a1614

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,       4;
  2:      2,      12;
  3:      6,      40,       9;
  4:     24,     168,      63;
  5:    120,     864,     405,     16;
  6:    720,    5280,    2808,    208;
  7:   5040,   37440,   21600,   2176,   25;
  8:  40320,  302400,  184680,  22400,  525;
  9: 362880, 2741760, 1746360, 240000, 8125, 36;
%F Formula
The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*n-8)*x*P(n-3,x) + (4*n-8)*x^2*d/dx P(n-3,x) - (n^2-3*n+2-4*x)*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 2 + 12*x, P(3,x) = 6 + 40*x + 9*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 2 + 12*x;
P[3, x_] := 6 + 40*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*n - 8)*x*P[n - 3, x] + (4*n - 8)*x^2*D[P[n - 3, x], x] - (n^2 - 3*n + 2 - 4*x)*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (2*n - 2)*P[n - 1, x]];
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] = 1 + 4*x; v[3] = 2 + 12*x; v[4] = 6 + 40*x + 9*x^2; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*n - 8)*x*v[i - 3] + (4*n - 8)*x^2*deriv(v[i - 3], x) - (n^2 - 3*n + 2 - 4*x)*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (2*n - 2)*v[i - 1]); 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.