Real-rooted atlas

Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.

A064861 (tail) 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 A064861
\(P_{n}(x) = \left(1 + 3\,x + 2\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1 + 3\,x + 2\,x^{2}\)
\(P_{1}(x) = 1 + 5\,x + 8\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01326
1158418
2161312436
3182538288108
419336366368216
5111511291921688016648
61126218032136024896161296
7114863046811002968592208323888

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-5a70a6e97131d437 · revision 1 · JSON
Definition source
oeis_catalog:A064861
Fingerprint
f6-9dc572a790c7fbc2

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 2)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1,  3,   2;
  1: 1,  5,   8,   4;
  2: 1,  6,  13,  12,    4;
  3: 1,  8,  25,  38,   28,    8;
  4: 1,  9,  33,  63,   66,   36,    8;
  5: 1, 11,  51, 129,  192,  168,   80,   16;
  6: 1, 12,  62, 180,  321,  360,  248,   96,   16;
  7: 1, 14,  86, 304,  681, 1002,  968,  592,  208,   32;
  8: 1, 15, 100, 390,  985, 1683, 1970, 1560,  800,  240,  32;
  9: 1, 17, 130, 590, 1765, 3653, 5336, 5500, 3920, 1840, 512, 64;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1 + 3*x + 2*x^2, P(1,x) = 1 + 5*x + 8*x^2 + 4*x^3.
%t Mathematica
P[0, x_] := 1 + 3*x + 2*x^2;
P[1, x_] := 1 + 5*x + 8*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x + 2*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x + 2*x^2; v[2] = 1 + 5*x + 8*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + 2*x^2)*v[i - 2]); 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-bbffc3f85cdc52e2
  • 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-bbffc3f85cdc52e2
  • 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-bbffc3f85cdc52e2

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 13 proof steps

Same fingerprint (2)

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