Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013610 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 12\,x + 3\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
228132144304
32801656302417286688
4364024888591845875220736167200
5582404418881245600165024010368002488324681600
611065609094752289690564630176039502080171694082985984145129600
72434432021336326474645625613662673921424666880851751936271724544358318084934406400

All 15 rows

Technical details
Node id
n-2a4db0762a391b72 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-28d339a36b0d6fb3

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,           12;
  2:          28,          132,          144;
  3:         280,         1656,         3024,          1728;
  4:        3640,        24888,        59184,         58752,         20736;
  5:       58240,       441888,      1245600,       1650240,       1036800,        248832;
  6:     1106560,      9094752,     28969056,      46301760,      39502080,      17169408,      2985984;
  7:    24344320,    213363264,    746456256,    1366267392,    1424666880,     851751936,    271724544,     35831808;
  8:   608608000,   5626213440,  21221765568,   43114159872,   52011880704,   38389800960,  17014136832,   4156489728,   429981696;
  9: 17041024000, 164837272320, 661723997184, 1461857663232, 1973702578176, 1699056995328, 937073442816, 320551354368, 61917364224, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+12*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 + 1 + 12*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 + 1 + 12*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