Real-rooted atlas

A128813 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 8\,n x - n^{2} - 4\,n^{2} x + 9\,n^{2} x^{2} - 3\,n^{3} - 4\,n^{3} x - 9\,n^{3} x^{2} + n^{4} + 4\,n^{4} x - 9\,n^{4} x^{2} - n^{5} - 4\,n^{5} x + 9\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n x^{2} + 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + 4\,n^{3} x^{2} - 4\,n^{3} x^{3} - 4\,n^{4} x^{2} - 4\,n^{4} x^{3} + 4\,n^{5} x^{2} + 4\,n^{5} x^{3}\right)\,P_{n-2}'(x) + \left(4\,n - 8\,n x + 4\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(8\,n x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
28402775
3564727292881545
461674321863916128450047315
59856155872549054755088450000972002017070
62168324257088194441403710995234676325157464002778300114229037
762881281477407368320766042044074688262917382518547950246806835001016064008296438905

All 15 rows

Technical details
Node id
n-c7f045471477b8b6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0b8a804eb8da2a31

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,            4;
  2:           8,           40,            27;
  3:          56,          472,           729,           288;
  4:         616,         7432,         18639,         16128,           4500;
  5:        9856,       155872,        549054,        755088,         450000,          97200;
  6:      216832,      4257088,      19444140,      37109952,       34676325,       15746400,        2778300;
  7:     6288128,    147740736,     832076604,    2044074688,     2629173825,     1854795024,      680683500,     101606400;
  8:   232660736,   6371897664,   42753833964,  128883666112,   212258632725,   204923786976,   116070245676,   35765452800,    4629441600;
  9: 10702393856, 334986225024, 2611831000824, 9348955358272, 18826029878850, 23180853601476, 17890813253376, 8467298737920, 2249908617600, 257191200000;
%F Formula
The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x-(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^5-4*n^4+4*n^3+4*n^2-8*n)*x^2+(4*n^5-4*n^4-4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(8*n^3-8*n)*x)*P(n-1,x) - (8*n^3-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*P[n - 1, x] - (8*n^3 - 8*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^5 - 4*n^4 + 4*n^3 + 4*n^2 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (8*n^3 - 8*n)*x)*v[i - 1] - (8*n^3 - 8*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.