Real-rooted atlas

A225470 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 36\,x + 81\,x^{2} + 18\,n - 36\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-36\,x^{2} + 36\,x^{3} + 36\,n x^{2}\right)\,P_{n-2}'(x) + \left(-3 + 12\,x + 6\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
21810881207
3162118814584323240
41944162002551512960202558644
5291602663284920753304809112587481217916
6524880514382410655064857304031893755511243572128673028
711022480114318864258477156236860848107163000253517043000564139968756334584

All 15 rows

Technical details
Node id
n-352bc8e38fcb0b30 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-12b23884bf72db83

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:          3,          12;
  2:         18,         108,           81;
  3:        162,        1188,         1458,          432;
  4:       1944,       16200,        25515,        12960,         2025;
  5:      29160,      266328,       492075,       330480,        91125,        8748;
  6:     524880,     5143824,     10655064,      8573040,      3189375,      551124,      35721;
  7:   11022480,   114318864,    258477156,    236860848,    107163000,    25351704,    3000564,     139968;
  8:  264539520,  2875922496,   6975104076,   7063205184,   3682197225,  1071385056,  175532994,   15116544,   531441;
  9: 7142567040, 80824381632, 207740286900, 227907095040, 132528099375, 44834488524, 9114213150, 1095949440, 71744535, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(36*n-36)*x-81*x^2)*P(n-2,x) + ((36*n-36)*x^2+36*x^3)*d/dx P(n-2,x) + (6*n-3+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (36*n - 36)*x - 81*x^2)*P[n - 2, x] + ((36*n - 36)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (6*n - 3 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], 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] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (36*n - 36)*x - 81*x^2)*v[i - 2] + ((36*n - 36)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (6*n - 3 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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. [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.