A204420 → 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

\(P_{n}(x) = \left(1 - 3\,x - 4\,n + 6\,n x + 4\,n^{2}\right)\,P_{n-1}(x)\)

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

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	9	36	27													72
3	225	1035	1215	405												2880
4	11025	55440	81270	45360	8505											201600
5	893025	4788315	8079750	5868450	1913625	229635										21772800
6	108056025	608855940	1135664145	976714200	425207475	90935460	7577955									3353011200
7	18261468225	107110838835	215672622165	209355601455	109951917075	31951184265	4827157335	295540245								697426329600
8	4108830350625	24921704808000	53346327734700	56810278324800	34160183407350	12136852728000	2523913692300	283718635200	13299311025							188305108992000
9	1187451971330625	7411923037393875	16688095660536300	19138833150336900	12769617199288950	5249719792166850	1348390546202700	210714283880100	18313151281425	678264862275						64023737057280000
10	428670161650355625	2743388978865034500	6446882146585055175	7860340219922190000	5700745298512514250	2623017025331703000	786003015332685150	152926117614270000	18621761793760125	1288703238322500	38661097149675					26761922089943040000
11	189043541287806830625	1236840759863452618875	3015908532312506505675	3872563612220544266025	3009230110499116754250	1515897461977569420750	511877402357611440150	116958607833852234450	17846542360747225125	1741489121107110375	98237847857324175	2435649120429525				13488008733331292160000
12	100004033341249813400625	667332766316625106698000	1680757626023894172204450	2256683839594230865618800	1859089617697250317353975	1009546635010573279620000	375380070723628741871100	97190644306783021394400	17510964849371086268175	2152659167957219922000	172130570872915104450	8066869886862586800	168059789309637225			8065829222532112711680000
13	62502520838281133375390625	424583281448484427691296875	1100523473738680740630131250	1536484221698186353927083750	1331182299030348763267644375	770398368208902073564048125	310328541828060959640937500	88897657996011544011832500	18233651353865655522189375	2658734343676093921363125	269031044392363434431250	17951586494757749583750	710052609833217275625	12604484198222791875		5646080455772478898176000000
14	45564337691106946230659765625	314583916363845919590362062500	836672858152825498562360728125	1209239398990810992003884685000	1094887117950677343090206533125	669446176645747861452870277500	288631774817577507536931335625	89943004567165353315541830000	20493042134644997940634486875	3415143096202990565971057500	411481113199796551330794375	34878221150459837635485000	1971706858643793110214375	66702930376995014602500	1020963220056046141875	4573325169175707907522560000000

Computed in your browser ?

Technical details

Node id: n-77311ba2b23da2eb · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fe2eac7fb802bf9f

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 ?

A327882: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:                9,               36,                27;
  3:              225,             1035,              1215,               405;
  4:            11025,            55440,             81270,             45360,              8505;
  5:           893025,          4788315,           8079750,           5868450,           1913625,           229635;
  6:        108056025,        608855940,        1135664145,         976714200,         425207475,         90935460,          7577955;
  7:      18261468225,     107110838835,      215672622165,      209355601455,      109951917075,      31951184265,       4827157335,       295540245;
  8:    4108830350625,   24921704808000,    53346327734700,    56810278324800,    34160183407350,   12136852728000,    2523913692300,    283718635200,    13299311025;
  9: 1187451971330625, 7411923037393875, 16688095660536300, 19138833150336900, 12769617199288950, 5249719792166850, 1348390546202700, 210714283880100, 18313151281425, 678264862275;

%F Formula

The row polynomials satisfy P(n,x) = (4*n^2-4*n+1+(6*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[(4*n^2 - 4*n + 1 + (6*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] = (4*n^2 - 4*n + 1 + (6*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-afc618911e5721d2
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-afc618911e5721d2
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-afc618911e5721d2

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-afc618911e5721d2

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers root_domain e-8239b1c6b4764045 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.