Real-rooted atlas

A128813 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 24\,n x - n^{2} - 12\,n^{2} x + 81\,n^{2} x^{2} - 3\,n^{3} - 12\,n^{3} x - 81\,n^{3} x^{2} + n^{4} + 12\,n^{4} x - 81\,n^{4} x^{2} - n^{5} - 12\,n^{5} x + 81\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-24\,n x^{2} + 12\,n^{2} x^{2} + 36\,n^{2} x^{3} + 12\,n^{3} x^{2} - 36\,n^{3} x^{3} - 12\,n^{4} x^{2} - 36\,n^{4} x^{3} + 12\,n^{5} x^{2} + 36\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(4\,n - 24\,n x + 4\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(24\,n x^{2} - 24\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
121214
28120243371
35614166561777615809
461622296167751435456364500990619
59856467616494148620387376364500002361960085875934
62168321277126417499726010019687042808782325382637520020253807009850492285
762881284432222087488689436551900165762129630798254507151908324962182715002222131968001445237955305

All 15 rows

Technical details
Node id
n-bbcc67c907ed26d9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cde818948dd99780

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:           2,            12;
  2:           8,           120,            243;
  3:          56,          1416,           6561,            7776;
  4:         616,         22296,         167751,          435456,           364500;
  5:        9856,        467616,        4941486,        20387376,         36450000,         23619600;
  6:      216832,      12771264,      174997260,      1001968704,       2808782325,       3826375200,        2025380700;
  7:     6288128,     443222208,     7488689436,     55190016576,     212963079825,     450715190832,      496218271500,      222213196800;
  8:   232660736,   19115692992,   384784505676,   3479858985024,   17192949250725,   49796480235168,    84615209097804,    78219045273600,    30373766337600;
  9: 10702393856, 1004958675072, 23506479007416, 252421794673344, 1524908420186850, 5632947425158668, 13042402861711104, 18517982339831040, 14761650440073600, 5062294389600000;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x-(81*n^5-81*n^4-81*n^3+81*n^2)*x^2)*P(n-2,x) + ((12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x^2+(36*n^5-36*n^4-36*n^3+36*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(24*n^3-24*n)*x)*P(n-1,x) - (24*n^3-24*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*P[n - 2, x] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*P[n - 1, x] - (24*n^3 - 24*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];
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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*v[i - 2] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*v[i - 1] - (24*n^3 - 24*n)*x^2*deriv(v[i - 1], x))/(4*n - 4)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.