Real-rooted atlas

A225470 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
2706816154
39101164480642618
4145602226412336294425652360
52766404812563234401052801664010241204280
66086080116941929040704360992078720089088409631311280
715215200031669912027279436812641081634119680537600045875216384908027120

All 15 rows

Technical details
Node id
n-89fc672875fe82c0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-76942a2a8b06805d

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:            7,            4;
  2:           70,           68,           16;
  3:          910,         1164,          480,           64;
  4:        14560,        22264,        12336,         2944,         256;
  5:       276640,       481256,       323440,       105280,       16640,        1024;
  6:      6086080,     11694192,      9040704,      3609920,      787200,       89088,       4096;
  7:    152152000,    316699120,    272794368,    126410816,    34119680,     5376000,     458752,     16384;
  8:   4260256000,   9476183360,   8905038784,   4630680320,  1460994304,   287006720,   34349056,   2293760,    65536;
  9: 132067936000, 310802708160, 313960935744, 179171245056, 63813544704, 14741185536, 2212847616, 208502784, 11206656, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+4+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 + 4 + 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 + 4 + 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.