Real-rooted atlas

A359363 → reverse

palindromicγ-positivenonnegative coefficients

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

Recurrence

\(\left(6 + 5\,n + n^{2}\right)\,P_{n}(x) = \left(6 + 5\,n - 10\,n x + n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(21\,x + 15\,x^{2} + 3\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2112
31416
411010122
51205020192
6135175175351422
71564909804905612074

All 15 rows

Technical details
Node id
n-978bdc86d94d4dda · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-509399d8a74ba170

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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: 1;
  2: 1,   1;
  3: 1,   4,    1;
  4: 1,  10,   10,     1;
  5: 1,  20,   50,    20,     1;
  6: 1,  35,  175,   175,    35,     1;
  7: 1,  56,  490,   980,   490,    56,    1;
  8: 1,  84, 1176,  4116,  4116,  1176,   84,   1;
  9: 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1;
%F Formula
The row polynomials satisfy (n^2+5*n+6)*P(n,x) = (n^2+5*n+6+(10*n^2-10*n)*x)*P(n-1,x) + ((3*n+21)*x-(15*n-15)*x^2)*d/dx P(n-1,x) + (6*x^2+6*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 5*n + 6 + (10*n^2 - 10*n)*x)*P[n - 1, x] + ((3*n + 21)*x - (15*n - 15)*x^2)*D[P[n - 1, x], x] + (6*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^2 + 5*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 5*n + 6 + (10*n^2 - 10*n)*x)*v[i - 1] + ((3*n + 21)*x - (15*n - 15)*x^2)*deriv(v[i - 1], x) + (6*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^2 + 5*n + 6)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.