A269947 → 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

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	3							4
2	2	9	9						20
3	18	87	108	27					240
4	504	2490	3285	1080	81				7440
5	32760	163362	220995	80055	8505	243			505920
6	4127760	20681892	28335456	10749915	1311795	56133	729		65263680
7	895723920	4500353844	6210839628	2417737923	316909260	16116246	326592	2187	14358009600

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-c2c5d2e7e07878ae · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-487c30882cf44b69

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): 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:               1;
  1:               1,               3;
  2:               2,               9,                9;
  3:              18,              87,              108,              27;
  4:             504,            2490,             3285,            1080,             81;
  5:           32760,          163362,           220995,           80055,           8505,           243;
  6:         4127760,        20681892,         28335456,        10749915,        1311795,         56133,          729;
  7:       895723920,      4500353844,       6210839628,      2417737923,      316909260,      16116246,       326592,       2187;
  8:    308129028480,   1550808894096,    2150029893564,    850334364396,   116269999209,    6494716404,    160696386,    1732104,    6561;
  9: 158070191610240, 796489349756688, 1107617762080620, 442671618615840, 62197512687405, 3680599512879, 101921395230, 1370658510, 8562105, 19683;

%F Formula

The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+3*x)*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[(n^3 - 3*n^2 + 3*n + 3*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; for(n = 1, N, my(i = n + 1); v[i] = (n^3 - 3*n^2 + 3*n + 3*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-f13e01bf2e1e97ea
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-f13e01bf2e1e97ea
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-f13e01bf2e1e97ea

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-f13e01bf2e1e97ea

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-f7365f1dbad216ac root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.