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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	308129028480	1550808894096	2150029893564	850334364396	116269999209	6494716404	160696386	1732104	6561							4982229331200
9	158070191610240	796489349756688	1107617762080620	442671618615840	62197512687405	3680599512879	101921395230	1370658510	8562105	19683						2570830334899200
10	115391239875475200	581911435897212960	810950434368122664	326473134875805060	46732199117653170	2873430182463885	85444417056537	1306344897990	10362312180	40054905	59049					1884418635481113600
11	115506631115350675200	582839521052736598560	813507119110182425544	329232459313785233052	47758350721398238350	3016500209999308395	94150152020985192	1563984494057601	14291709186150	71181896445	179272764	177147				1891956310023038054400
12	153854832645647099366400	776688761935591201307520	1085340001217921200620288	440978157163292477701896	64601820538843809181356	4161253331883273497190	134457503121950200929	2365677802147680108	23728510118124603	137689413623190	452337010983	773778096	531441			2525761673880755802624000
13	266015005644323834804505600	1343356433884574128358801280	1878882928391592529476400512	765707253738986457548439048	113019482183150823507670212	7388612472442711304185578	244960782893501717897811	4493629429279189509519	48123627400680478911	309250526508869319	1195158932859177	2694873360933	3240195777	1594323		4374619219161469050144768000
14	584700982406223788900303308800	2953495486695226905637058730240	4133814745906374102174204729216	1688661192503467011279898229040	250713943599782469442504443120	16579228660978531917122911080	560589638217244909851945312	10611879834236163695616195	119256621314533261174935	824103539468536199895	3554710913951079003	9508808445908265	15206570400645	13224909285	4782969	9628536901374393379368634368000

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.