A204420 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	1														4
2	27	36	9													72
3	405	1215	1035	225												2880
4	8505	45360	81270	55440	11025											201600
5	229635	1913625	5868450	8079750	4788315	893025										21772800
6	7577955	90935460	425207475	976714200	1135664145	608855940	108056025									3353011200
7	295540245	4827157335	31951184265	109951917075	209355601455	215672622165	107110838835	18261468225								697426329600
8	13299311025	283718635200	2523913692300	12136852728000	34160183407350	56810278324800	53346327734700	24921704808000	4108830350625							188305108992000
9	678264862275	18313151281425	210714283880100	1348390546202700	5249719792166850	12769617199288950	19138833150336900	16688095660536300	7411923037393875	1187451971330625						64023737057280000
10	38661097149675	1288703238322500	18621761793760125	152926117614270000	786003015332685150	2623017025331703000	5700745298512514250	7860340219922190000	6446882146585055175	2743388978865034500	428670161650355625					26761922089943040000
11	2435649120429525	98237847857324175	1741489121107110375	17846542360747225125	116958607833852234450	511877402357611440150	1515897461977569420750	3009230110499116754250	3872563612220544266025	3015908532312506505675	1236840759863452618875	189043541287806830625				13488008733331292160000
12	168059789309637225	8066869886862586800	172130570872915104450	2152659167957219922000	17510964849371086268175	97190644306783021394400	375380070723628741871100	1009546635010573279620000	1859089617697250317353975	2256683839594230865618800	1680757626023894172204450	667332766316625106698000	100004033341249813400625			8065829222532112711680000
13	12604484198222791875	710052609833217275625	17951586494757749583750	269031044392363434431250	2658734343676093921363125	18233651353865655522189375	88897657996011544011832500	310328541828060959640937500	770398368208902073564048125	1331182299030348763267644375	1536484221698186353927083750	1100523473738680740630131250	424583281448484427691296875	62502520838281133375390625		5646080455772478898176000000
14	1020963220056046141875	66702930376995014602500	1971706858643793110214375	34878221150459837635485000	411481113199796551330794375	3415143096202990565971057500	20493042134644997940634486875	89943004567165353315541830000	288631774817577507536931335625	669446176645747861452870277500	1094887117950677343090206533125	1209239398990810992003884685000	836672858152825498562360728125	314583916363845919590362062500	45564337691106946230659765625	4573325169175707907522560000000

Technical details

Node id: n-4643327b1b00e955 · revision 0 · JSON
Definition source: derived:reversal
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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            1;
  1:            3,              1;
  2:           27,             36,               9;
  3:          405,           1215,            1035,              225;
  4:         8505,          45360,           81270,            55440,            11025;
  5:       229635,        1913625,         5868450,          8079750,          4788315,            893025;
  6:      7577955,       90935460,       425207475,        976714200,       1135664145,         608855940,         108056025;
  7:    295540245,     4827157335,     31951184265,     109951917075,     209355601455,      215672622165,      107110838835,       18261468225;
  8:  13299311025,   283718635200,   2523913692300,   12136852728000,   34160183407350,    56810278324800,    53346327734700,    24921704808000,    4108830350625;
  9: 678264862275, 18313151281425, 210714283880100, 1348390546202700, 5249719792166850, 12769617199288950, 19138833150336900, 16688095660536300, 7411923037393875, 1187451971330625;

%F Formula

The row polynomials satisfy P(n,x) = (6*n-3+(4*n^2-4*n+1)*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[(6*n - 3 + (4*n^2 - 4*n + 1)*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] = (6*n - 3 + (4*n^2 - 4*n + 1)*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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of e…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A204420 → basis A013610

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.