Real-rooted atlas

A257623 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 5\,n\right)\,P_{n}(x) = \left(18 - 84\,x - 42\,x^{2} + 30\,n - 80\,n x + 65\,n x^{2} + 100\,n^{2} x + 150\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 30\,x^{3} + 50\,n x - 75\,n x^{2} + 25\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-12 + 8\,x + 5\,n - 40\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-10\,x + 10\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
299627132
327100215031082640
4818988390781797640566528
524375630750690100092018907514582018016
67296186481236820536033120206136751855944510371495424
7218749972981867698811027887780128485972537353976217490543174962895564672

All 15 rows

Technical details
Node id
n-214a184f3ede7fe7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ade7ffc8995f784b

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,        96,          27;
  3:    27,      1002,        1503,          108;
  4:    81,      8988,       39078,        17976,           405;
  5:   243,     75630,      750690,      1000920,        189075,          1458;
  6:   729,    618648,    12368205,     36033120,      20613675,       1855944,          5103;
  7:  2187,   4997298,   186769881,   1027887780,    1284859725,     373539762,      17490543,       17496;
  8:  6561,  40144596,  2675374704,  25474722264,   59103547350,   38212083396,    6242540976,   160578384,      59049;
  9: 19683, 321721014, 37068113124, 576261487728, 2250996868290, 2701196241948, 1008457603524, 98848301664, 1447744563, 196830;
%F Formula
The row polynomials satisfy (5*n-2)*P(n,x) = (30*n+18+(100*n^2-80*n-84)*x-(125*n^3-150*n^2-65*n+42)*x^2)*P(n-2,x) + ((50*n+30)*x-(125*n^2+75*n)*x^2+(125*n^2+25*n-30)*x^3)*d/dx P(n-2,x) + (5*n-12+(50*n^2-40*n+8)*x)*P(n-1,x) + ((25*n-10)*x-(25*n-10)*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[((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*P[n - 2, x] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*D[P[n - 2, x], x] + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*P[n - 1, x] + ((25*n - 10)*x - (25*n - 10)*x^2)*D[P[n - 1, x], x])/(5*n - 2)]];
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] = ((30*n + 18 + (100*n^2 - 80*n - 84)*x - (125*n^3 - 150*n^2 - 65*n + 42)*x^2)*v[i - 2] + ((50*n + 30)*x - (125*n^2 + 75*n)*x^2 + (125*n^2 + 25*n - 30)*x^3)*deriv(v[i - 2], x) + (5*n - 12 + (50*n^2 - 40*n + 8)*x)*v[i - 1] + ((25*n - 10)*x - (25*n - 10)*x^2)*deriv(v[i - 1], x))/(5*n - 2)); 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.