Real-rooted atlas

A286718 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 4\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2546016130
3648936432642080
497201663210224268825639520
517496033825625056089280153601024869440
6367416078032166614784287712067968082944409621736000
7881798402019738241899676809551001627820800470937643008016384608608000

All 15 rows

Technical details
Node id
n-b4ecca4f366c3049 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7dba1ab6c8440df9

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:           6,            4;
  2:          54,           60,           16;
  3:         648,          936,          432,           64;
  4:        9720,        16632,        10224,         2688,         256;
  5:      174960,       338256,       250560,        89280,       15360,        1024;
  6:     3674160,      7803216,      6614784,      2877120,      679680,       82944,       4096;
  7:    88179840,    201973824,    189967680,     95510016,    27820800,     4709376,     430080,     16384;
  8:  2380855680,   5806012608,   5937022656,   3338641152,  1133201664,   238436352,   30449664,   2162688,    65536;
  9: 71425670400, 183703800960, 201334730112, 123907325184, 47350614528, 11685897216, 1867235328, 186679296, 10616832, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3+4*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 + 3 + 4*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 + 3 + 4*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.