Real-rooted atlas

basis-map coefficients of A371080, shifted → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-2 - 2\,x + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
24181840
328162288162640
4280195648245076194414080
53640296289205213834810141229160394240
658240539568200613638705044112856229197652488013404160
7110656011474832494475121156684321594248481299175205810421611022480536166400
82434432027900374413632412323731445792628338902466843817924396313232163699574426453952024663654400
96086080007632390240416141318881300936580642578337619843367610484482903861391841596253508645081237308871425670401282510028800
101704102400023196516672013941674000644891046382432111221550774721716432221606418233643350592131810939997126211506972384172436306976021427701120074385581670400
11528271744000775327396032050874039903744197629962057504506191338021888899125106254560113166557699846410103241445606086275328181344002584349852512326354656864928070711413696004760677226905600
121796123929600028262909743488020088352192988168550884146489920243251841268143364879314178144300870845133443111936750909816870053765770778502075148831377970951380288114642428031198722528095277940480254561089305600333247405883392000
1366456585395200011157764937634560853494379140165123947272869727808641233516294405237312275402842685915040045242024668714189445541326526700564416506373633156101472034115885410103786881647917850819266496540644994605472768108014476143985920992788248291840025326802847137792000
142658263415808000047422236337136640038826036439413120001937376787129992806465919197829066288768161968821439385983104296637292602941074560411669564676622172288435285167383864294272349140467565977765760209203432755206564736908383495186281035522702766881918929305649337232973641446404169710642825728002076797833465298944000
Technical details
Node id
n-637206203e2bde41 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-863fd2e33b5cca56

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:         1,          3;
  2:         4,         18,          18;
  3:        28,        162,         288,          162;
  4:       280,       1956,        4824,         5076,         1944;
  5:      3640,      29628,       92052,       138348,       101412,        29160;
  6:     58240,     539568,     2006136,      3870504,      4112856,      2291976,       524880;
  7:   1106560,   11474832,    49447512,    115668432,    159424848,    129917520,     58104216,     11022480;
  8:  24344320,  279003744,  1363241232,   3731445792,   6283389024,   6684381792,   4396313232,   1636995744,   264539520;
  9: 608608000, 7632390240, 41614131888, 130093658064, 257833761984, 336761048448, 290386139184, 159625350864, 50812373088, 7142567040;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n-2+(6*n-2)*x+3*n*x^2)*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[Together[((3*n - 2 + (6*n - 2)*x + 3*n*x^2)*P[n - 1, x])/(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 - 2 + (6*n - 2)*x + 3*n*x^2)*v[i - 1])/(1 + x)); 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