Real-rooted atlas

A034870 → basis A008277

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(4 + 5\,x + x^{2}\right)\,P_{n-1}(x) + \left(5\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
145110
2166555141151
36466513519102452713263
42566305263353557420181541871444194828
5102458025465751113267011441655634091471472112016506513535027
640965273457859215325665345411506143613856190127084775628713427636353289901163254885
7163844766585129442951885423630231423328528731412711925136213753655760181092340274574302650648161070591511919097183602

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-f8e3d642daa0223a · 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:      4,         5,           1;
  2:     16,        65,          55,           14,             1;
  3:     64,       665,        1351,          910,           245,            27,              1;
  4:    256,      6305,       26335,        35574,         20181,          5418,            714,             44,             1;
  5:   1024,     58025,      465751,      1132670,       1144165,        563409,         147147,          21120,          1650,            65,            1;
  6:   4096,    527345,     7859215,     32566534,      54115061,      43613856,       19012708,        4775628,        713427,         63635,         3289,          90,          1;
  7:  16384,   4766585,   129442951,    885423630,    2314233285,    2873141271,     1925136213,      753655760,     181092340,      27457430,      2650648,      161070,       5915,       119,        1;
  8:  65536,  42981185,  2104469695,  23305343894,   93181501141,  171754378614,   168620069982,    96646573452,   34353829653,    7878943930,   1194306542,   120944460,    8158878,    359380,     9860,    152,     1;
  9: 262144, 387158345, 33972448951, 601616805790, 3612997293605, 9650629410813, 13461181659199, 10866668017920, 5440287930870, 1771429211695, 387549682091, 58176221220, 6058947050, 438412422, 21823818, 728688, 15504, 189, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+5*x+x^2)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + 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[(4 + 5*x + x^2)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + 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] = (4 + 5*x + x^2)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + 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-d4da514254176df5
  • 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-d4da514254176df5
  • 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-d4da514254176df5

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