Real-rooted atlas

A113214 → basis A013609

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-2}(x) + \left(1 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2410418
372628869
411581047216261
51812030034417632990
62923676012241024416643753
747450177236884336284896012814229
8768383904997615184139847552217625653946
91231534825225016468485587242176193284864512204525
101992770169005930413201619392018944012096048128107521024775413
113224948337601346243475046075207312006035203335681172482355220482939814
12521876066096295248867376176048025235202565760183065689139228057651200409611145681
13843153941272726295842074624479776079907209678720850278453370882321408661504110592819242256485
14136426882241676131156847916641244224023630720332729603482240026895360150589445916672154009623756816384160206498

Computed in your browser ?

Technical details
Node id
n-30551c14ec291558 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-9a10453378e11448

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:   3,    2;
  2:   4,   10,    4;
  3:   7,   26,   28,     8;
  4:  11,   58,  104,    72,    16;
  5:  18,  120,  300,   344,   176,    32;
  6:  29,  236,  760,  1224,  1024,   416,    64;
  7:  47,  450, 1772,  3688,  4336,  2848,   960,   128;
  8:  76,  838, 3904,  9976, 15184, 13984,  7552,  2176,  256;
  9: 123, 1534, 8252, 25016, 46848, 55872, 42176, 19328, 4864, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (1 + 2*x)*P[n - 1, 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] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (1 + 2*x)*v[i - 1]); 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-ba56344dd1f68fd7
  • 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-ba56344dd1f68fd7
  • 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-ba56344dd1f68fd7

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