Real-rooted atlas

A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.

A200536 nonnegative coefficients only rational roots

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

Recurrence from OEIS A200536
\(P_{n}(x) = \left(1 + 3\,x + 2\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11326
2161312436
319336366368216
41126218032136024896161296
5115100390985168319701560800240327776
6118147720235554188989108369420576023525766446656
7121203119748091392329953486395990655692384721915264961344128279936

Show all 15 stored rows

Generated b-files: A000400 (row sums), A200536 ?

Computed in your browser ?

Technical details
Node id
n-eed4f744ce063213 · revision 1 · JSON
Definition source
oeis_catalog:A200536
Fingerprint
f6-99cef2a9f1688781

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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 1,  3,   2;
  2: 1,  6,  13,   12,     4;
  3: 1,  9,  33,   63,    66,    36,      8;
  4: 1, 12,  62,  180,   321,   360,    248,     96,     16;
  5: 1, 15, 100,  390,   985,  1683,   1970,   1560,    800,     240,      32;
  6: 1, 18, 147,  720,  2355,  5418,   8989,  10836,   9420,    5760,    2352,     576,      64;
  7: 1, 21, 203, 1197,  4809, 13923,  29953,  48639,  59906,   55692,   38472,   19152,    6496,   1344,    128;
  8: 1, 24, 268, 1848,  8806, 30744,  81340, 166344, 265729,  332688,  325360,  245952,  140896,  59136,  17152,   3072,   256;
  9: 1, 27, 342, 2700, 14886, 60858, 191184, 471852, 927441, 1462563, 1854882, 1887408, 1529472, 973728, 476352, 172800, 43776, 6912, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+2*x^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[(1 + 3*x + 2*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x + 2*x^2)*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.

A200536: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  3,   2;
  2: 1,  6,  13,   12,     4;
  3: 1,  9,  33,   63,    66,    36,      8;
  4: 1, 12,  62,  180,   321,   360,    248,     96,     16;
  5: 1, 15, 100,  390,   985,  1683,   1970,   1560,    800,     240,      32;
  6: 1, 18, 147,  720,  2355,  5418,   8989,  10836,   9420,    5760,    2352,     576,      64;
  7: 1, 21, 203, 1197,  4809, 13923,  29953,  48639,  59906,   55692,   38472,   19152,    6496,   1344,    128;
  8: 1, 24, 268, 1848,  8806, 30744,  81340, 166344, 265729,  332688,  325360,  245952,  140896,  59136,  17152,   3072,   256;
  9: 1, 27, 342, 2700, 14886, 60858, 191184, 471852, 927441, 1462563, 1854882, 1887408, 1529472, 973728, 476352, 172800, 43776, 6912, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+2*x^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[(1 + 3*x + 2*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x + 2*x^2)*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-982ad459e6d62a8e
  • 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-982ad459e6d62a8e
  • 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-982ad459e6d62a8e

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

Used in 5 proof steps

No candidate equalities or related sequences.