Real-rooted atlas

A130534 → D²∘x (shift +1) → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 3\,x^{2} - x^{3} + 3\,n x - 6\,n x^{2} + 3\,n x^{3} + 3\,n^{2} x^{2} - 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n-3}(x) + \left(-3 + 3\,x - x^{2} - 6\,n x + 3\,n x^{2} - 3\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 6 + 18\,x + 11\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
26181135
3106010550225
4151505106752741624
521315175044104872176413132
628588483019600406143939613068118124
736100811466680402244904037043543721095841172700

All 15 rows

Technical details
Node id
n-69f0f1363c9e6eb5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-86ccdfaf0e2feef5

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,    3;
  2:  6,   18,    11;
  3: 10,   60,   105,     50;
  4: 15,  150,   510,    675,     274;
  5: 21,  315,  1750,   4410,    4872,     1764;
  6: 28,  588,  4830,  19600,   40614,    39396,    13068;
  7: 36, 1008, 11466,  68040,  224490,   403704,   354372,   109584;
  8: 45, 1620, 24360, 198450,  949095,  2693250,  4342080,  3518100,  1026576;
  9: 55, 2475, 47520, 508200, 3313233, 13530825, 34169300, 50457000, 38260728, 10628640;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n-3)*x+(3*n^2-6*n+3)*x^2+(n^3-3*n^2+3*n-1)*x^3)*P(n-3,x) - (3+(6*n-3)*x+(3*n^2-3*n+1)*x^2)*P(n-2,x) + (3+3*n*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 6 + 18*x + 11*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 6 + 18*x + 11*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n - 3)*x + (3*n^2 - 6*n + 3)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*P[n - 3, x] - (3 + (6*n - 3)*x + (3*n^2 - 3*n + 1)*x^2)*P[n - 2, x] + (3 + 3*n*x)*P[n - 1, x]];
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 + 3*x; v[3] = 6 + 18*x + 11*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + (3*n - 3)*x + (3*n^2 - 6*n + 3)*x^2 + (n^3 - 3*n^2 + 3*n - 1)*x^3)*v[i - 3] - (3 + (6*n - 3)*x + (3*n^2 - 3*n + 1)*x^2)*v[i - 2] + (3 + 3*n*x)*v[i - 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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.