Real-rooted atlas

A256890 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 20 + 20\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
12020444
2120180768384
384016801100260163896
46720168001500057008443245096
56048018144020580010920027020266064586664
660480021168002916480199920070812012138082361288475144
7665280026611200431827203640896016952040426888052670025220256134628776
879833600359251200671328000673142400390479040131594400245046002232900766845122332443336

All 15 rows

Technical details
Node id
n-a62fc9bae1fba01f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-cef044b741d89456

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          20,          20,            4;
   2:         120,         180,           76,            8;
   3:         840,        1680,         1100,          260,           16;
   4:        6720,       16800,        15000,         5700,          844,           32;
   5:       60480,      181440,       205800,       109200,        27020,         2660,          64;
   6:      604800,     2116800,      2916480,      1999200,       708120,       121380,        8236,        128;
   7:     6652800,    26611200,     43182720,     36408960,     16952040,      4268880,      526700,      25220,       256;
   8:    79833600,   359251200,    671328000,    673142400,    390479040,    131594400,    24504600,    2232900,     76684,      512;
   9:  1037836800,  5189184000,  10977120000,  12773376000,   8899450560,   3786108480,   961098600,  135920400,   9315020,   232100,   1024;
  10: 14529715200, 79913433600, 188806464000, 250278336000, 203954889600, 105270580800, 34191548160, 6718034400, 735492120, 38420580, 700396, 2048;
%F Formula
The row polynomials satisfy P(n,x) = (n+4+(n+2)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 20 + 20*x + 4*x^2.
%t Mathematica
P[1, x_] := 20 + 20*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + (n + 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 20 + 20*x + 4*x^2; for(n = 2, N, my(i = n); v[i] = (n + 4 + (n + 2)*x)*v[i - 1] - (x + 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, v[i])))};
T(10)

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