Real-rooted atlas

A059364 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
22830866
3280468260481056
4364083247124270438422176
558240169584197224114504331843840576576
6110656039209765782264454226420045444711684608017873856
724344320101753312182103472180881504107691664384293127610112645120643458816

All 15 rows

Technical details
Node id
n-42c5fee8e0c9f6a5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5f874879722b2208

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,           2;
  2:          28,          30,            8;
  3:         280,         468,          260,           48;
  4:        3640,        8324,         7124,         2704,          384;
  5:       58240,      169584,       197224,       114504,        33184,         3840;
  6:     1106560,     3920976,      5782264,      4542264,      2004544,       471168,       46080;
  7:    24344320,   101753312,    182103472,    180881504,    107691664,     38429312,     7610112,      645120;
  8:   608608000,  2933341920,   6180639792,   7435693152,   5586395664,   2683799424,   805121792,   137889792,   10321920;
  9: 17041024000, 93088517760, 225858068736, 319450924512, 290261555328, 175701505824, 70851799808, 18353106432, 2771030016, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+2*n*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 + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 1 + 2*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

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.