Real-rooted atlas

A204420 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
22418345
3720660180151575
4403204200014700210010599225
5362880041428801701000321300283509459823275
6479001600586776960270103680611226007276500436590103951404728325
787178291200113020427520567869702401463566104021189168001740538807567560135135273922023375

All 15 rows

Technical details
Node id
n-6251a83089056710 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-181e40cb14f29d52

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:                2,                1;
  2:               24,               18,                3;
  3:              720,              660,              180,               15;
  4:            40320,            42000,            14700,             2100,             105;
  5:          3628800,          4142880,          1701000,           321300,           28350,            945;
  6:        479001600,        586776960,        270103680,         61122600,         7276500,         436590,         10395;
  7:      87178291200,     113020427520,      56786970240,      14635661040,      2118916800,      174053880,       7567560,       135135;
  8:   20922789888000,   28432576972800,   15324179270400,    4364363203200,    728074947600,    73556683200,    4427022600,    145945800,    2027025;
  9: 6402373705728000, 9056055981772800, 5172552665280000, 1596006187776000, 296985108420000, 34885619168400, 2605132530000, 119918799000, 3101348250, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-2*n+(2*n-1)*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*n + (2*n - 1)*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*n + (2*n - 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.