Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013609 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
2410418
38487828162
4161766368362801944
5325603560994011428364029160
6641632160807684018189619012858240524880
7128448063168459200182375238362803728912110656011022480

All 15 rows

Technical details
Node id
n-6f17df1b336821a1 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d1d0e40f5b436a84

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:   4,    10,      4;
  3:   8,    48,     78,       28;
  4:  16,   176,    636,      836,      280;
  5:  32,   560,   3560,     9940,    11428,      3640;
  6:  64,  1632,  16080,    76840,   181896,    190128,      58240;
  7: 128,  4480,  63168,   459200,  1823752,   3836280,    3728912,    1106560;
  8: 256, 11776, 224896,  2308096, 13749904,  47795104,   91855984,   84249184,   24344320;
  9: 512, 29952, 744192, 10238592, 85202208, 439337808, 1378589568, 2464897968, 2154918240, 608608000;
%F Formula
The row polynomials satisfy P(n,x) = (2+(3*n-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[(2 + (3*n - 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] = (2 + (3*n - 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 ?