Real-rooted atlas

A143494 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2415
3189128
49067151173
54944851632211165
62946357115953253018468
7189262718115091411057539165923
8130066215195142387484619100938491546197
9950654177558913602835544141306411813014426014791214
10735379415269731132637716286309178084231111533390211872144301143
115995463813675753313257021171514670236441954981557674667578403000851430158397
1251333361812740231471361266819822326461310483800765742121253798713589079537041259914372004546
13460138076612328516629143755828279601422446407193898111501075522209787802901188425775971509755533114146381674085
14430625563221237462952191561905482671141610517975367800981417074652309376606172858135105662660598465093223094572671301512328076385
15419742815726128655622548517458558956031384741989358714331653399252542621331628741843731116349291214169950081277416148044036343070937314715879362011436
Technical details
Node id
n-4d43c95e2ed5d558 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-20bd10e5918a2042

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 -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:       1;
   2:       4,        1;
   3:      18,        9,        1;
   4:      90,       67,       15,       1;
   5:     494,      485,      163,      22,       1;
   6:    2946,     3571,     1595,     325,      30,      1;
   7:   18926,    27181,    15091,    4110,     575,     39,     1;
   8:  130066,   215195,   142387,   48461,    9100,    938,    49,    1;
   9:  950654,  1775589,  1360283,  554414,  130641,  18130,  1442,   60,  1;
  10: 7353794, 15269731, 13263771, 6286309, 1780842, 311115, 33390, 2118, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (2+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + x)*P[n - 1, x] + (2 + x)*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] = 1; for(n = 2, N, my(i = n); v[i] = (4 + x)*v[i - 1] + (2 + x)*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.