Real-rooted atlas

A099040 → basis A008277

nonnegative coefficients

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2614424
31680568160
444420548184161212
51202120462427445443210184
63281048836112337681145615046493720
789651328269632373568193024424323968128934976
824482497121958096387017628594729361921443201011225610030784
96688121084813968448384054083901244817848480403328046041625088512114971616
1018272586051298480384370057760503270368309159904960557441588364813972486092810241400245792
1149920283353606888357123492603008623975449650086662402055263104461683072582901764073472145408204818037702016
12136384136919872479228832032475918336751555701767729996345640754116160119437052162029507072202062848114903043420164096244802024256
133726086613935363321725465629871041881688589028940811504942056967639485690882841049711366239112704082983208966682583043152896079462481923488417512960
14101798431942716162296538846722725509783168102728312701441665562174899213721425931776634618419212817516687667202988489431043195523686421244006408450867218268161638452039105799680

Computed in your browser ?

Technical details
Node id
n-f4f4a67a77e0ecc6 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-04b957434080eed3

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 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:    2,       2;
  2:    6,      14,        4;
  3:   16,      80,       56,        8;
  4:   44,     420,      548,      184,       16;
  5:  120,    2120,     4624,     2744,      544,       32;
  6:  328,   10488,    36112,    33768,    11456,     1504,      64;
  7:  896,   51328,   269632,   373568,   193024,    42432,    3968,    128;
  8: 2448,  249712,  1958096,  3870176,  2859472,   936192,  144320,  10112,   256;
  9: 6688, 1210848, 13968448, 38405408, 39012448, 17848480, 4033280, 460416, 25088, 512;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (2+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (2 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], x) + (2 + 2*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 - 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-a2850f8f2b718783
  • 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-a2850f8f2b718783
  • 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-a2850f8f2b718783

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