Real-rooted atlas

A225471 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 2\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
26032496
384056812081536
4151201190432963841630720
53326402921289632015040112032737280
68648640826060830885765836805920030726420643840
72594592002651155201091784962368755229433602105608064128660602880

All 15 rows

Technical details
Node id
n-d3e87c9d56ff1da1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e2f382b41485ddb6

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,            2;
  2:           60,           32,            4;
  3:          840,          568,          120,           8;
  4:        15120,        11904,         3296,         384,         16;
  5:       332640,       292128,        96320,       15040,       1120,        32;
  6:      8648640,      8260608,      3088576,      583680,      59200,      3072,       64;
  7:    259459200,    265115520,    109178496,    23687552,    2943360,    210560,     8064,     128;
  8:   8821612800,   9532846080,   4242299904,  1023733760,  147449344,  13045760,   695296,   20480,   256;
  9: 335221286400, 379891376640, 180273088512, 47386482688, 7650542592, 790637568, 52512768, 2168832, 50688, 512;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+2+2*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[(4*n + 2 + 2*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] = (4*n + 2 + 2*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

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.