Real-rooted atlas

A257626 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(9 + 3\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-3\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112618
218018036396
332404860205221610368
46804013608089100210601296315576
51632960408240036450001385100205092777610958328
6440899201322697601500282007960680019697580193914046656427678056
7132269760046294416006378341760437225040015486584402654580601801213227993618535139928

All 15 rows

Technical details
Node id
n-b706353c3a058005 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cef044b741d89456

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:            12,             6;
  2:           180,           180,             36;
  3:          3240,          4860,           2052,            216;
  4:         68040,        136080,          89100,          21060,          1296;
  5:       1632960,       4082400,        3645000,        1385100,        205092,          7776;
  6:      44089920,     132269760,      150028200,       79606800,      19697580,       1939140,        46656;
  7:    1322697600,    4629441600,     6378341760,     4372250400,    1548658440,     265458060,     18012132,      279936;
  8:   43649020800,  174596083200,   283321825920,   238879186560,  111222334440,   28008121680,   3455678700,   165468420,    1679616;
  9: 1571364748800, 7071141369600, 13213749024000, 13249461859200, 7685798944320, 2590172575200, 482324041800, 43950170700, 1509371172, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+9+(3*n+3)*x)*P(n-1,x) - (3*x+3*x^2)*d/dx 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 + 9 + (3*n + 3)*x)*P[n - 1, x] - (3*x + 3*x^2)*D[P[n - 1, x], 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 + 9 + (3*n + 3)*x)*v[i - 1] - (3*x + 3*x^2)*deriv(v[i - 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 ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003