Real-rooted atlas

A200536 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(6 + 9\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(9\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
169217
236189185524466
32163159770958631726204818885
41296493292573514012062524577344010328672161046095
57776751599784969722527895256021901379687738740345905204856020003275051180
646656113439692295778551148703322217852972919406263479161978932457864363882370836340001959525568646713471499
7279936170579439657574484955617160963168364887586233357647446170578970681717211078711830299843829227989002952601681867924871120014784128727926841637

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-e529baa18a6a4626 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-f5044f2210a14593

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,           9,             2;
  2:       36,         189,           185,              52,               4;
  3:      216,        3159,          7709,            5863,            1726,              204,                8;
  4:     1296,       49329,        257351,          401206,          252457,            73440,            10328,              672,               16;
  5:     7776,      751599,       7849697,        22527895,        25602190,         13796877,          3874034,           590520,            48560,            2000,              32;
  6:    46656,    11343969,     229577855,      1148703322,      2178529729,       1940626347,        916197893,        245786436,         38823708,         3634000,          195952,           5568,            64;
  7:   279936,   170579439,    6575744849,     55617160963,    168364887586,     233357647446,     170578970681,      71721107871,      18302998438,      2922798900,       295260168,       18679248,        711200,        14784,         128;
  8:  1679616,  2561211009,  186338948351,   2614901064646,  12293194450597,   25555902429162,   27573578032826,   17069150831628,    6472408635809,   1569031697600,    249623753744,    26384101632,    1849354080,     84327936,     2387712,     37888,     256;
  9: 10077696, 38433281679, 5250789656177, 120843432453055, 866636433993670, 2638146784447959, 4074536923286012, 3572619957180666, 1916463377639598, 661953377877025, 152447399696962, 23944460359056, 2596660736224, 194864037984, 10027017408, 345112320, 7550976, 94464, 512;
%F Formula
The row polynomials satisfy P(n,x) = (6+9*x+2*x^2)*P(n-1,x) + (9*x+4*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^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[(6 + 9*x + 2*x^2)*P[n - 1, x] + (9*x + 4*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}]];
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] = (6 + 9*x + 2*x^2)*v[i - 1] + (9*x + 4*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), 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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-4c5ee4490c6f16fe
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-4c5ee4490c6f16fe
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-4c5ee4490c6f16fe

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [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