Real-rooted atlas

A120910 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,x + 18\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(27 + 54\,x + 36\,x^{2} + 8\,x^{3} + 54\,n x + 60\,n x^{2} + 16\,n x^{3} + 12\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
192433
227144108279
3816489723842085
4243259258324608120014475
57299720291603456018000345695625
6218734992131220207360162000622089408609375
7656112247255112410886401134000653184197568245763778125

All 15 rows

Technical details
Node id
n-cdb87b58d20a5654 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e543af4f498eab8b

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:     3;
  1:     9,      24;
  2:    27,     144,     108;
  3:    81,     648,     972,      384;
  4:   243,    2592,    5832,     4608,     1200;
  5:   729,    9720,   29160,    34560,    18000,     3456;
  6:  2187,   34992,  131220,   207360,   162000,    62208,     9408;
  7:  6561,  122472,  551124,  1088640,  1134000,   653184,   197568,   24576;
  8: 19683,  419904, 2204496,  5225472,  6804000,  5225472,  2370816,  589824,   62208;
  9: 59049, 1417176, 8503056, 23514624, 36741600, 35271936, 21337344, 7962624, 1679616, 153600;
%F Formula
The row polynomials satisfy (9+(18*n-6)*x+4*n^2*x^2)*P(n,x) = (27+(54*n+54)*x+(12*n^2+60*n+36)*x^2+(8*n^2+16*n+8)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 3.
%t Mathematica
P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(9 + (18*n - 6)*x + 4*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((27 + (54*n + 54)*x + (12*n^2 + 60*n + 36)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(9 + (18*n - 6)*x + 4*n^2*x^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

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.