Real-rooted atlas

A376827 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-12 - 15\,x - 3\,x^{2} + 12\,n + 15\,n x + 3\,n x^{2}\right)\,P_{n-2}(x) + \left(-15\,x - 6\,x^{2} + 15\,n x + 6\,n x^{2}\right)\,P_{n-2}'(x) + \left(-3\,x^{2} + 3\,n x^{2}\right)\,P_{n-2}''(x) + \left(-2 - x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21215330
356133637259
430412351045266192869
516321076114535637510205134374
6902493765190491128310345453807141460083
75030480918723851172319093935340170562137553936683751

All 15 rows

Technical details
Node id
n-d168bf213caff51d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5f1389370a379f1

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,        1;
  2:      12,       15,         3;
  3:      56,      133,        63,         7;
  4:     304,     1235,      1045,       266,        19;
  5:    1632,    10761,     14535,      6375,      1020,        51;
  6:    9024,    93765,    190491,    128310,     34545,      3807,      141;
  7:   50304,   809187,   2385117,   2319093,    935340,    170562,    13755,     393;
  8:  283392,  6979635,  29152845,  39380418,  22340367,   5997726,   790398,   48708,   1107;
  9: 1607168, 60177769, 350453655, 640999495, 491755740, 182397873, 34937070, 3484290, 169506, 3139;
%F Formula
The row polynomials satisfy n*P(n,x) = (12*n-12+(15*n-15)*x+(3*n-3)*x^2)*P(n-2,x) + ((15*n-15)*x+(6*n-6)*x^2)*d/dx P(n-2,x) + (3*n-3)*x^2*d^2/dx^2 P(n-2,x) + (4*n-2+(2*n-1)*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n - 12 + (15*n - 15)*x + (3*n - 3)*x^2)*P[n - 2, x] + ((15*n - 15)*x + (6*n - 6)*x^2)*D[P[n - 2, x], x] + (3*n - 3)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x])/(n)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n - 12 + (15*n - 15)*x + (3*n - 3)*x^2)*v[i - 2] + ((15*n - 15)*x + (6*n - 6)*x^2)*deriv(v[i - 2], x) + (3*n - 3)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 2 + (2*n - 1)*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x))/(n)); 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.