Real-rooted atlas

A257621 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,n\right)\,P_{n}(x) = \left(9 - 30\,x - 15\,x^{2} + 12\,n - 16\,n x + 52\,n x^{2} + 32\,n^{2} x + 48\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 12\,x^{3} + 16\,n x - 48\,n x^{2} + 32\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-6 + 2\,x + 8\,n - 16\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x + 4\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
298427120
32778611791082100
4816312259381262440545360
52434726242733856978411815514581164240
6729342012604768517093520100794751026036510334594560
721872433450783239494096108205120135251566478988517075174961167566400

All 15 rows

Technical details
Node id
n-40ab481764296b44 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ca6524666417859c

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,         6;
  2:     9,        84,          27;
  3:    27,       786,        1179,          108;
  4:    81,      6312,       25938,        12624,          405;
  5:   243,     47262,      427338,       569784,       118155,         1458;
  6:   729,    342012,     6047685,     17093520,     10079475,      1026036,         5103;
  7:  2187,   2433450,    78323949,    409610820,    512013525,    156647898,      8517075,       17496;
  8:  6561,  17169744,   960118164,   8546096736,  19456513950,  12819145104,   2240275716,    68678976,     59049;
  9: 19683, 120647478, 11359692900, 162755704944, 615374046210, 738448855452, 284822483652, 30292514400, 542913651, 196830;
%F Formula
The row polynomials satisfy (4*n-1)*P(n,x) = (12*n+9+(32*n^2-16*n-30)*x-(64*n^3-48*n^2-52*n+15)*x^2)*P(n-2,x) + ((16*n+12)*x-(64*n^2+48*n)*x^2+(64*n^2+32*n-12)*x^3)*d/dx P(n-2,x) + (8*n-6+(32*n^2-16*n+2)*x)*P(n-1,x) + ((16*n-4)*x-(16*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*P[n - 2, x] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*D[P[n - 2, x], x] + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*P[n - 1, x] + ((16*n - 4)*x - (16*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n - 1)]];
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 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*v[i - 2] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*deriv(v[i - 2], x) + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*v[i - 1] + ((16*n - 4)*x - (16*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n - 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.