A064861 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 21\,x + 18\,x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 6 + 21\,x + 18\,x^{2}\)

\(P_{1}(x) = 18 + 99\,x + 180\,x^{2} + 108\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	21	18								45
1	18	99	180	108							405
2	36	252	657	756	324						2025
3	108	972	3483	6210	5508	1944					18225
4	216	2268	9882	22869	29646	20412	5832				91125
5	648	8100	43254	127899	226152	239112	139968	34992			820125
6	1296	18144	110808	385560	836001	1156680	997272	489888	104976		4100625
7	3888	62208	441288	1821528	4821363	8486046	9931896	7453296	3254256	629856	36905625

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0d95f7d056b9311c · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2e6f63c0c32e8505

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 ?

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:     6,     21,      18;
  1:    18,     99,     180,      108;
  2:    36,    252,     657,      756,      324;
  3:   108,    972,    3483,     6210,     5508,      1944;
  4:   216,   2268,    9882,    22869,    29646,     20412,      5832;
  5:   648,   8100,   43254,   127899,   226152,    239112,    139968,     34992;
  6:  1296,  18144,  110808,   385560,   836001,   1156680,    997272,    489888,    104976;
  7:  3888,  62208,  441288,  1821528,  4821363,   8486046,   9931896,   7453296,   3254256,    629856;
  8:  7776, 136080, 1069200,  4966920, 15107310,  31436181,  45321930,  44702280,  28868400,  11022480,  1889568;
  9: 23328, 454896, 4024080, 21315960, 75123450, 184952403, 324582876, 406038420, 354818880, 206277840, 71803584, 11337408;

%F Formula

The row polynomials satisfy P(n,x) = (6+21*x+18*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 6 + 21*x + 18*x^2, P(1,x) = 18 + 99*x + 180*x^2 + 108*x^3.

%t Mathematica

P[0, x_] := 6 + 21*x + 18*x^2;
P[1, x_] := 18 + 99*x + 180*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[(6 + 21*x + 18*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] = 6 + 21*x + 18*x^2; v[2] = 18 + 99*x + 180*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = (6 + 21*x + 18*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-f5362ff16cbef54d
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-f5362ff16cbef54d
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-f5362ff16cbef54d

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

basis_map A013610 e-f5362ff16cbef54d

Geometric basis map (elementary) polytool ?
- premise (sequence): 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.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-bbffc3f85cdc52e2 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.