Real-rooted atlas

basis-map coefficients of A112227 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence
\(P_{n}(x) = \left(1 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1181625
21124864125
311696256256625
4120160640128010243125
5124240128038406144409615625
61283362240896021504286721638478125
71324483584179205734411468813107265536390625
81365765376322561290243440645898245898242621441953125

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-146f1cd501c36740 · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-03d7b8417365c509

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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): yes

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,  8,  16;
   2: 1, 12,  48,    64;
   3: 1, 16,  96,   256,   256;
   4: 1, 20, 160,   640,  1280,   1024;
   5: 1, 24, 240,  1280,  3840,   6144,    4096;
   6: 1, 28, 336,  2240,  8960,  21504,   28672,   16384;
   7: 1, 32, 448,  3584, 17920,  57344,  114688,  131072,    65536;
   8: 1, 36, 576,  5376, 32256, 129024,  344064,  589824,   589824,   262144;
   9: 1, 40, 720,  7680, 53760, 258048,  860160, 1966080,  2949120,  2621440,  1048576;
  10: 1, 44, 880, 10560, 84480, 473088, 1892352, 5406720, 10813440, 14417920, 11534336, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 16*x^2.
%t Mathematica
P[1, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-b91bf4c52fd06745
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-b91bf4c52fd06745
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-b91bf4c52fd06745

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 (14)

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