Real-rooted atlas

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

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2915428
32710811728280
481594143112542803640
52432835120152236517142364058240
67291239381405259335409266285192582401106560
72187510304796822324700615516386316305593368110656024344320
865612012042561706175271046960888916130847620667596412637377624344320608608000
91968376763712715218116623962647004267222414765346527397925546020428323237736060860800017041024000
1059049285403559639490705897990520648373724788562435762343536601469147754601649857040649233239008017041024000528271744000
1117714710392624267393555396651816037502288901235766683152997148496465280400928984050493151514525391553996224291342716448052827174400017961239296000
12531441372008701155529881209909353502473684841431982377872090110075127165634231507674933011048426130891618785137713804019205311736505610064133882432017961239296000664565853952000
13159432313126592748430218331057274116471518770060379150997675295611063705194170195342232007608211897110623651958465147179984401275266603062026487407889358980032377761325438784066456585395200026582634158080000
14478296945757070119779702579510903108261878540664701710605010500384392310225943349758574903789632232706025990138871489838840345610356208638852912368424323290080325046016307648414122364800153098227737369600265826341580800001143053268797440000
Technical details
Node id
n-b5ff02f9a0149f78 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f061f7c708547767

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:     3,      1;
  2:     9,     15,        4;
  3:    27,    108,      117,        28;
  4:    81,    594,     1431,      1254,       280;
  5:   243,   2835,    12015,     22365,     17142,       3640;
  6:   729,  12393,    81405,    259335,    409266,     285192,      58240;
  7:  2187,  51030,   479682,   2324700,   6155163,    8631630,    5593368,    1106560;
  8:  6561, 201204,  2561706,  17527104,  69608889,  161308476,  206675964,  126373776,   24344320;
  9: 19683, 767637, 12715218, 116623962, 647004267, 2224147653, 4652739792, 5546020428, 3232377360, 608608000;
%F Formula
The row polynomials satisfy P(n,x) = (3+(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[(3 + (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] = (3 + (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 ?