Real-rooted atlas

A156366 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27\,x^{2} - 36\,n x^{2} - 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-9\,x^{2} + 27\,x^{3} - 3\,n x^{2} + 9\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 + 12\,x + 6\,n x\right)\,P_{n-1}(x) + \left(x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 24\,x + 27\,x^{2}\)
\(P_{1}(x) = 1 + 66\,x + 297\,x^{2} + 108\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01242752
1166297108472
21156178228084055152
3134281543261623085145865656
41720321572609284823551749605103956224
51148211591116868526325695625919412604411749615667072
61301239441695292726325695012864517274544624878299259049285215488
71607812916804916073653069337019104961322322844776837008640598166371968305711514880

All 15 rows

Technical details
Node id
n-9016ac5e146d9d18 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-bb5ae5a9769469ab

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,    24,       27;
  1: 1,    66,      297,        108;
  2: 1,   156,     1782,       2808,         405;
  3: 1,   342,     8154,      32616,       23085,         1458;
  4: 1,   720,    32157,     260928,      482355,       174960,         5103;
  5: 1,  1482,   115911,    1686852,     6325695,      6259194,      1260441,         17496;
  6: 1,  3012,   394416,    9529272,    63256950,    128645172,     74544624,       8782992,        59049;
  7: 1,  6078,  1291680,   49160736,   530693370,   1910496132,   2322844776,     837008640,     59816637,      196830;
  8: 1, 12216,  4121199,  237976704,  3943936170,  22925953584,  49693595742,   38552226048,   9013062213,   400745880,     649539;
  9: 1, 24498, 12913317, 1100269980, 26858981970, 236942912988, 829300195458, 1160308021104, 601572611565, 94138080930, 2652067737, 2125764;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2+36*n+27)*x^2*P(n-2,x) - ((3*n+9)*x^2-(9*n+27)*x^3)*d/dx P(n-2,x) + (1+(6*n+12)*x)*P(n-1,x) + (x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 24*x + 27*x^2, P(1,x) = 1 + 66*x + 297*x^2 + 108*x^3.
%t Mathematica
P[0, x_] := 1 + 24*x + 27*x^2;
P[1, x_] := 1 + 66*x + 297*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 + 36*n + 27)*x^2*P[n - 2, x] - ((3*n + 9)*x^2 - (9*n + 27)*x^3)*D[P[n - 2, x], x] + (1 + (6*n + 12)*x)*P[n - 1, x] + (x - 3*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 + 24*x + 27*x^2; v[2] = 1 + 66*x + 297*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 + 36*n + 27)*x^2*v[i - 2] - ((3*n + 9)*x^2 - (9*n + 27)*x^3)*deriv(v[i - 2], x) + (1 + (6*n + 12)*x)*v[i - 1] + (x - 3*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

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.