Real-rooted atlas

A156366 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 6\,x + 3\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-2 - 5\,x - 3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 22 + 30\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02230961
116031218027679
21456369631861026819445
3159044975257312290525832243158095
42026727535521087776757296253206335347293088765
5295168012715392220734721959206492586242169504195372218768958295
64836121623685427248057292851917414431847904010875902418518058115036265611731875605
788040550448311093761121977036814293463040108053766724881245976125196710415850501268234401968348328686175

All 15 rows

Technical details
Node id
n-eab480e9d0419e9c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-62dec3872b40fd4c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           22,            30,             9;
  1:          160,           312,           180,             27;
  2:         1456,          3696,          3186,           1026,             81;
  3:        15904,         49752,         57312,          29052,           5832,           243;
  4:       202672,        753552,       1087776,         757296,         253206,         33534,           729;
  5:      2951680,      12715392,      22073472,       19592064,        9258624,       2169504,        195372,          2187;
  6:     48361216,     236854272,     480572928,      519174144,      318479040,     108759024,      18518058,       1150362,         6561;
  7:    880405504,    4831109376,   11219770368,    14293463040,    10805376672,    4881245976,    1251967104,     158505012,      6823440,       19683;
  8:  17630351872,  107141927424,  280294352640,   411526884096,   370209407472,  208760650704,   72251586576,   14280410160,   1364779854,    40665078,     59049;
  9: 385148108800, 2567740096512, 7472160981504, 12425283059328, 12971970415872, 8786224374336, 3846772316736, 1047656910816, 162444271392, 11834521848, 243045684, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+7+(3*n+6)*x)*P(n-1,x) - (2+5*x+3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 22 + 30*x + 9*x^2.
%t Mathematica
P[0, x_] := 22 + 30*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(3*n + 7 + (3*n + 6)*x)*P[n - 1, x] - (2 + 5*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] = 22 + 30*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 7 + (3*n + 6)*x)*v[i - 1] - (2 + 5*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

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