Real-rooted atlas

A120909 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1271845
2135324108567
3567243029166486561
42187136082916023328388872171
580196561020412029160017496023328767637
62843128868411809802449440262440012597121399687971615
79841511941026062364165337202571912022044960881798483980881310473

All 15 rows

Technical details
Node id
n-7920008a31ba6c4f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       3;
  1:      27,       18;
  2:     135,      324,       108;
  3:     567,     2430,      2916,       648;
  4:    2187,    13608,     29160,     23328,       3888;
  5:    8019,    65610,    204120,    291600,     174960,      23328;
  6:   28431,   288684,   1180980,   2449440,    2624400,    1259712,     139968;
  7:   98415,  1194102,   6062364,  16533720,   25719120,   22044960,    8817984,     839808;
  8:  334611,  4723920,  28658448,  96997824,  198404640,  246903552,  176359680,   60466176,   5038848;
  9: 1121931, 18068994, 127545840, 515852064, 1309470624, 2142770112, 2222131968, 1360488960, 408146688, 30233088;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (6*n+3+(12*n+12)*x+12*x^2)*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[((6*n + 3 + (12*n + 12)*x + 12*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];
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] = ((6*n + 3 + (12*n + 12)*x + 12*x^2)*v[i - 1])/(2*n - 1 + 2*x)); 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.