Real-rooted atlas

A139548 → basis A008277

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(8 + 10\,x + 2\,x^{2}\right)\,P_{n-1}(x) + \left(10\,x + 4\,x^{2}\right)\,P_{n-1}'(x) + \left(2\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1810220
264260220564604
351253201080872801960216826104
440961008804213605691843228968668811424704161517248
532768185680014904032362454403661328018029088470870467584052800208032113120864
626214433750080502989760208425817634633639042791286784121681331230564019245659328407264021049657606410448312640
72097152610122880165686977281133342246402962218604803677620826882464174352649646793728023179819520351455104033928294420616960757120152321281164439501056

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f2ca92e21d86d85e · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-910fa4df8c34a164

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:         8,           10,              2;
  2:        64,          260,            220,              56,                4;
  3:       512,         5320,          10808,            7280,             1960,              216,                8;
  4:      4096,       100880,         421360,          569184,           322896,            86688,            11424,              704,               16;
  5:     32768,      1856800,       14904032,        36245440,         36613280,         18029088,          4708704,           675840,            52800,            2080,              32;
  6:    262144,     33750080,      502989760,      2084258176,       3463363904,       2791286784,       1216813312,        305640192,         45659328,         4072640,          210496,           5760,            64;
  7:   2097152,    610122880,    16568697728,    113334224640,     296221860480,     367762082688,     246417435264,      96467937280,      23179819520,      3514551040,       339282944,       20616960,        757120,        15232,         128;
  8:  16777216,  11003183360,   538744241920,   5966168036864,   23854464292096,   43969120925184,   43166737915392,   24741522803712,    8794580391168,   2017009646080,    305742474752,    30961781760,    2088672768,     92001280,     2524160,     38912,     256;
  9: 134217728, 198225072640, 17393893862912, 308027804564480, 1849854614325760, 4941122258336256, 6892125009509888, 5563734025175040, 2785427420605440, 906971756387840, 198425437230592, 29786225264640, 3102180889600, 224467160064, 11173794816, 373088256, 7938048, 96768, 512;
%F Formula
The row polynomials satisfy P(n,x) = (8+10*x+2*x^2)*P(n-1,x) + (10*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[(8 + 10*x + 2*x^2)*P[n - 1, x] + (10*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] = (8 + 10*x + 2*x^2)*v[i - 1] + (10*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-b4af3e745574d56e
  • 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-b4af3e745574d56e
  • 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-b4af3e745574d56e

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

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