A269947 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

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

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

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	1														2
2	2	3	1													6
3	18	29	12	1												60
4	504	830	365	40	1											1740
5	32760	54454	24555	2965	105	1										114840
6	4127760	6893964	3148384	398145	16195	231	1									14584680
7	895723920	1500117948	690093292	89545849	3912460	66322	448	1								3179460240
8	308129028480	516936298032	238892210396	31493865348	1435432089	26727228	220434	792	1							1096913782800
9	158070191610240	265496449918896	123068640231180	16395245133920	767870527005	15146500053	139809870	626730	1305	1						563813684359200
10	115391239875475200	193970478632404320	90105603818680296	12091597587992780	576940729847570	11824815565695	117207705153	597322770	1579380	2035	1					412147803266575200
11	115506631115350675200	194279840350912199520	90389679901131380616	12193794789399453076	589609268165410350	12413581111108265	129149728423848	715127797923	2178282150	3616415	3036	1				412972098873108350400
12	153854832645647099366400	258896253978530400435840	120593333468657911180032	16332524339381202877848	797553339985726039276	17124499308161619330	184441019371673801	1081699955257284	3616599621723	6995346930	7660367	4368	1			550491807797853431083200
13	266015005644323834804505600	447785477961524709452933760	208764769821288058830711168	28359527916258757686979224	1395302249174701524786052	30405812643797165860846	336023021801785621259	2054700242011517837	7334800701216351	15711554463693	20240121473	15212639	6097	1		952350827490286435773936000
14	584700982406223788900303308800	984498495565075635212352910080	459312749545152678019356081024	62543007129758037454811045520	3095233871602252709166721520	68227278440240872086925560	768984414564121961388128	4852254153743101826985	18176592183285057335	41868797412413565	60199341461347	53677501995	28613845	8295	1	2094219469651139872266885264000

Computed in your browser ?

Technical details

Node id: n-b096a5459ca672a4 · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-759e2feadaf031a1

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,               1;
  2:               2,               3,               1;
  3:              18,              29,              12,              1;
  4:             504,             830,             365,             40,            1;
  5:           32760,           54454,           24555,           2965,          105,           1;
  6:         4127760,         6893964,         3148384,         398145,        16195,         231,         1;
  7:       895723920,      1500117948,       690093292,       89545849,      3912460,       66322,       448,      1;
  8:    308129028480,    516936298032,    238892210396,    31493865348,   1435432089,    26727228,    220434,    792,    1;
  9: 158070191610240, 265496449918896, 123068640231180, 16395245133920, 767870527005, 15146500053, 139809870, 626730, 1305, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+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 + 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 + 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-0e122e2df114c72b
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-0e122e2df114c72b
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-0e122e2df114c72b

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 A007318 e-0e122e2df114c72b

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.