A269947 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(4\,x + 3\,n - 3\,n^{2} + n^{3}\right)\,P_{n-1}(x)\)

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

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	4														5
2	2	12	16													30
3	18	116	192	64												390
4	504	3320	5840	2560	256											12480
5	32760	217816	392880	189760	26880	1024										861120
6	4127760	27575856	50374144	25481280	4145920	236544	4096									111945600
7	895723920	6000471792	11041492672	5730934336	1001589760	67913728	1835008	16384								24739977600
8	308129028480	2067745192128	3822275366336	2015607382272	367470614784	27368681472	902897664	12976128	65536							8609512204800
9	158070191610240	1061985799675584	1969098243698880	1049295688570880	196574854913280	15510016054272	572661227520	10268344320	85524480	262144						4451117809881600
10	115391239875475200	775881914529617280	1441689661098884736	773862245631537920	147696826840977920	12108611139271680	480082760306688	9786536263680	103506247680	533463040	1048576					3267120472453094400
11	115506631115350675200	777119361403648798080	1446234878418102089856	780402866521564996864	150939972650345049600	12711507057774863360	528997287624081408	11716653841170432	142755898982400	948021493760	3183476736	4194304				3283456074815359872000
12	153854832645647099366400	1035585015914121601743360	1929493335498526578880512	1045281557720396984182272	204173655036345866054656	17535487291557498193920	755470415346375888896	17722572066935341056	237017472809238528	1833788225617920	8032476987392	18320719872	16777216			4386697315953320788992000
13	266015005644323834804505600	1791141911846098837811735040	3340236317140608941291378688	1815009786640560491966670336	357197375788723590345229312	31135552147248297841506304	1376350297300113904676864	33664208765116708241408	480693498754914779136	4118689733330337792	21223305613672448	63806432608256	102290685952	67108864		7602146448547104927323136000
14	584700982406223788900303308800	3937993982260302540849411640320	7349003992722442848309697296384	4002752456304514397107906913280	792379871130176693546680709120	69864733122806653017011773440	3149760162054643553845772288	79499332054926980333322240	1191221145323769517506560	10975654028879741583360	63123584672173391872	225139761327636480	480060658155520	556668026880	268435456	16739926479700725049965545472000

Computed in your browser ?

Technical details

Node id: n-2a7a2754f632f0f9 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e3473847634e4d8f

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,                4;
  2:               2,               12,               16;
  3:              18,              116,              192,               64;
  4:             504,             3320,             5840,             2560,             256;
  5:           32760,           217816,           392880,           189760,           26880,           1024;
  6:         4127760,         27575856,         50374144,         25481280,         4145920,         236544,         4096;
  7:       895723920,       6000471792,      11041492672,       5730934336,      1001589760,       67913728,      1835008,       16384;
  8:    308129028480,    2067745192128,    3822275366336,    2015607382272,    367470614784,    27368681472,    902897664,    12976128,    65536;
  9: 158070191610240, 1061985799675584, 1969098243698880, 1049295688570880, 196574854913280, 15510016054272, 572661227520, 10268344320, 85524480, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n+4*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 + 4*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 + 4*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-8aecb1bbda67a980
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-8aecb1bbda67a980
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-8aecb1bbda67a980

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 A013611 e-8aecb1bbda67a980

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.