Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
22833970
328041418927910
43640622236999188114560
55824011047277850257854050243276640
6110656022736881810566723465154305167677296086080
7243443205334081646653516213479285565105831789663392187152152000

All 15 rows

Technical details
Node id
n-52bdb0b8dcc3da52 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4e282d90b4560864

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:           1;
  1:           4,           3;
  2:          28,          33,           9;
  3:         280,         414,         189,          27;
  4:        3640,        6222,        3699,         918,         81;
  5:       58240,      110472,       77850,       25785,       4050,        243;
  6:     1106560,     2273688,     1810566,      723465,     154305,      16767,       729;
  7:    24344320,    53340816,    46653516,    21347928,    5565105,     831789,     66339,     2187;
  8:   608608000,  1406553360,  1326360348,   673658748,  203171409,   37490040,   4153842,   253692,   6561;
  9: 17041024000, 41209318080, 41357749824, 22841525988, 7709775696, 1659235347, 228777696, 19564902, 944784, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+3*x)*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[(3*n + 1 + 3*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 1 + 3*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

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.