Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 - 2\,x + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
21620440
36419215628440
42561408254416722806160
5102489602848039760228563640104720
6409652224257280614720727584380256582402094400
7163842867202021376734720014590016153451207457824110656048171200

All 15 rows

Technical details
Node id
n-69138fbe889402d3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-0c88df3e4716fde5

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:      4,       1;
  2:     16,      20,        4;
  3:     64,     192,      156,        28;
  4:    256,    1408,     2544,      1672,        280;
  5:   1024,    8960,    28480,     39760,      22856,       3640;
  6:   4096,   52224,   257280,    614720,     727584,     380256,       58240;
  7:  16384,  286720,  2021376,   7347200,   14590016,   15345120,     7457824,    1106560;
  8:  65536, 1507328, 14393344,  73859072,  219998464,  382360832,   367423936,  168498368,   24344320;
  9: 262144, 7667712, 95256576, 655269888, 2726470656, 7029404928, 11028716544, 9859591872, 4309836480, 608608000;
%F Formula
The row polynomials satisfy P(n,x) = (4+(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[(4 + (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] = (4 + (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 ?