A204420 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 54\,x - 27\,x^{2} + 108\,n - 180\,n x + 72\,n x^{2} - 144\,n^{2} + 168\,n^{2} x - 36\,n^{2} x^{2} + 80\,n^{3} - 48\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 6\,x - 12\,n + 12\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 6\,x\)

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	6														7
2	9	72	81													162
3	225	2070	3645	1620												7560
4	11025	110880	243810	181440	42525											589680
5	893025	9576630	24239250	23473800	9568125	1377810										69128640
6	108056025	1217711880	3406992435	3906856800	2126037375	545612760	53045685									11364312960
7	18261468225	214221677670	647017866495	837422405820	549759585375	191707105590	33790101345	2364321960								2494544532480
8	4108830350625	49843409616000	160038983204100	227241113299200	170800917036750	72821116368000	17667395846100	2269749081600	119693799225							704911208601600
9	1187451971330625	14823846074787750	50064286981608900	76555332601347600	63848085996444750	31498318753001100	9438733823418900	1685714271040800	164818361532825	6782648622750						249273371483136000
10	428670161650355625	5486777957730069000	19340646439755165525	31441360879688760000	28503726492562571250	15738102151990218000	5502021107328796050	1223408940914160000	167595856143841125	12887032383225000	425272068646425					107845622292215808000
11	189043541287806830625	2473681519726905237750	9047725596937519517025	15490254448882177064100	15046150552495583771250	9095384771865416524500	3583141816503280081050	935668862670817875600	160618881246725026125	17414891211071103750	1080616326430565925	29227789445154300				56040194726943178752000
12	100004033341249813400625	1334665532633250213396000	5042272878071682516613350	9026735358376923462475200	9295448088486251586769875	6057279810063439677720000	2627660495065401193097700	777525154454264171155200	157598683644339776413575	21526591679572199220000	1893436279602066148950	96802438642351041600	2184777261025283925			34442709049311880052736000
13	62502520838281133375390625	849166562896968855382593750	3301570421216042221890393750	6145936886792745415708335000	6655911495151743816338221875	4622390209253412441384288750	2172299792796426717486562500	711181263968092352094660000	164102862184790899699704375	26587343436760939213631250	2959341488315997778743750	215419037937092995005000	9230683927831824583125	176462778775119086250		24714833526208224490291200000
14	45564337691106946230659765625	629167832727691839180724125000	2510018574458476495687082184375	4836957595963243968015538740000	5474435589753386715451032665625	4016677059874487168717221665000	2020422423723042552758519349375	719544036537322826524334640000	184437379211804981465710381875	34151430962029905659710575000	4526292245197762064638738125	418538653805518051625820000	25632189162369310432786875	933841025277930204435000	15314448300840692128125	20476347673146232627888128000000

Technical details

Node id: n-321747bdedd90149 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e496ecd3ecc24bd1

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:                1;
  1:                1,                 6;
  2:                9,                72,                81;
  3:              225,              2070,              3645,              1620;
  4:            11025,            110880,            243810,            181440,             42525;
  5:           893025,           9576630,          24239250,          23473800,           9568125,           1377810;
  6:        108056025,        1217711880,        3406992435,        3906856800,        2126037375,         545612760,         53045685;
  7:      18261468225,      214221677670,      647017866495,      837422405820,      549759585375,      191707105590,      33790101345,       2364321960;
  8:    4108830350625,    49843409616000,   160038983204100,   227241113299200,   170800917036750,    72821116368000,   17667395846100,    2269749081600,    119693799225;
  9: 1187451971330625, 14823846074787750, 50064286981608900, 76555332601347600, 63848085996444750, 31498318753001100, 9438733823418900, 1685714271040800, 164818361532825, 6782648622750;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(48*n^3-168*n^2+180*n-54)*x+(36*n^2-72*n+27)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (12*n - 6)*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; v[2] = 1 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (48*n^3 - 168*n^2 + 180*n - 54)*x + (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (12*n - 6)*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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)

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.