Real-rooted atlas

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

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(-3 - 5\,x - 2\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1011
20347
3018452891
401625857022801729
50194491261602912486364043225
6029160167994386451443754254376582401339975
7052488035779321014800415330141130100945881464110656049579075
801102248086683932291822588545189049610475076409732812152629968243443202131900225
9026453952023559763689170840412203801018762828112804925095464388139064395324400012880608608000104463111025
100714256704071018468496313580029428807046282188133423330985114694491238481078146870228508180654656139632776640170410240005745471106375
1102142770112002351973633120116089734062163393236937790865045434043358854447063208217789732894612848667972616748199425835935364839846795840528271744000350473737488875
1207071141369600849005482737604630632259312081514473285282308330019988227968650312200660612655475731870130138425455228052103623128163275561208377627864429441819877586105601796123929600023481740411754625
130254561089305600331805196853056019811596419652608716543776296177841748427073175140923032313180225539223832814897689517733557659612935724022406798221712986521157336644315224243754878240836908873801516832467206645658539520001714167050058087625
14099278824829184001395864703449158409053743391076741123586984584341197896968504069056776094818819729695580166638270772308218912760272920613208120739459823617151516423543508141408057996813206166093749091187291392178977721198138560032112413563399680026582634158080000135419196954588922375
Technical details
Node id
n-42873d283a4afdf0 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-13e69f2906ee7e95

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: 0,         1;
  2: 0,         3,          4;
  3: 0,        18,         45,         28;
  4: 0,       162,        585,        702,         280;
  5: 0,      1944,       9126,      16029,       12486,        3640;
  6: 0,     29160,     167994,     386451,      443754,      254376,       58240;
  7: 0,    524880,    3577932,   10148004,    15330141,    13010094,     5881464,     1106560;
  8: 0,  11022480,   86683932,  291822588,   545189049,   610475076,   409732812,   152629968,   24344320;
  9: 0, 264539520, 2355976368, 9170840412, 20380101876, 28281128049, 25095464388, 13906439532, 4400012880, 608608000;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (3*n-3+(6*n-5)*x+(3*n-2)*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 - 3 + (6*n - 5)*x + (3*n - 2)*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 - 3 + (6*n - 5)*x + (3*n - 2)*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

Same fingerprint ?