Real-rooted atlas

A113214 → basis A013611

nonnegative coefficients

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

Recurrence
\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-2}(x) + \left(1 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
24201640
375211264235
4111164165762561375
51824012002752281610248050
629472304097921638413312409647125
74790070882950469376911366144016384275875
87616761561679808242944447488483328278528655361615000
912330683300820012874956817879042699264247398412451842621449454375
10199554067600474432211225662054401212416015482880123207685505024104857655346875
113229896135040107699255600641944064046796800772505608539340860030976241172484194304324006250
125211752026438423619841387801656335360161505280328417280468647936456392704287309824104857600167772161896765625
1384330788509088503667233193984153528320511406080123887616021767127042732589056237712179213547601924529848326710886411103859375
1413645376496670410492544766666243981516801512366080425893888089145344001377042432015420358656121173442566308233216194615705626843545665003125000

Computed in your browser ?

Technical details
Node id
n-7decb80581bfcd12 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-c39f98a1ebdd5382

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,    4;
  2:   4,   20,    16;
  3:   7,   52,   112,     64;
  4:  11,  116,   416,    576,    256;
  5:  18,  240,  1200,   2752,   2816,    1024;
  6:  29,  472,  3040,   9792,  16384,   13312,    4096;
  7:  47,  900,  7088,  29504,  69376,   91136,   61440,   16384;
  8:  76, 1676, 15616,  79808, 242944,  447488,  483328,  278528,   65536;
  9: 123, 3068, 33008, 200128, 749568, 1787904, 2699264, 2473984, 1245184, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x)*P(n-2,x) + (1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 2, x] + (1 + 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 2] + (1 + 4*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-ed3e6644af44bc4a
  • 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-ed3e6644af44bc4a
  • 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-ed3e6644af44bc4a

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