A204420 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 72\,x + 108\,x^{2} + 108\,n - 240\,n x - 288\,n x^{2} - 144\,n^{2} + 224\,n^{2} x + 144\,n^{2} x^{2} + 80\,n^{3} - 64\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(-72\,x^{2} + 48\,x^{3} + 240\,n x^{2} - 128\,n x^{3} - 224\,n^{2} x^{2} + 64\,n^{2} x^{3} + 64\,n^{3} x^{2}\right)\,P_{n-2}'(x) + \left(4 - 8\,x - 12\,n + 16\,n x + 8\,n^{2}\right)\,P_{n-1}(x) + \left(8\,x^{2} - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

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	8														9
2	9	96	108													213
3	225	2760	4860	1920												9765
4	11025	147840	325080	215040	42000											740985
5	893025	12768840	32319000	27820800	9450000	1088640										84340305
6	108056025	1623615840	4542656580	4630348800	2099790000	431101440	32598720									13468167405
7	18261468225	285628903560	862690488660	992500629120	542972430000	151472280960	20765384640	1107025920								2875398611085
8	4108830350625	66457879488000	213385310938800	269322800947200	168692263740000	57537672192000	10857329683200	1062744883200	42032390400							791466864613425
9	1187451971330625	19765128099717000	66752382642145200	90732246046041600	63059838021180000	24887560496198400	5800483885996800	789287340441600	57878601580800	1764322560000						273034021427192025
10	428670161650355625	7315703943640092000	25787528586340220700	37263835116668160000	28151828634629700000	12435043675646592000	3381214617398217600	572826134200320000	58853963200032000	3352212864000000	81123551308800					115398938169788998725
11	189043541287806830625	3298242026302540317000	12063634129250026022700	18358820087564061705600	14860395607403045700000	7186476856782551328000	2201985643063744137600	438100098479614771200	56403886967299872000	4530012853766400000	206134943875660800	4054836680294400				58657842079735013039925
12	100004033341249813400625	1779554043511000284528000	6723030504095576688817800	10698353017335612992563200	9180689470109878110390000	4785998862272347399680000	1614803706158161501382400	364053844609586882150400	55343296314061704748800	5599560503958220800000	361185900750862387200	13429619085135052800	218904863559782400			35307805172676133155683625
13	62502520838281133375390625	1132222083862625140510125000	4402093894954722962520525000	7284073347309920492691360000	6573739748298018584037750000	3652258930768128348748080000	1334966127809052358620000000	332990221446422756536320000	57627342550488985354080000	6915978173866585824000000	564514598458761350400000	29885565526607831040000	924873048540080640000	12693891496366080000		24839985533742452753845790625
14	45564337691106946230659765625	838890443636922452240965500000	3346691432611301994249442912500	5732690484104585443573971840000	5406849965188530089334353250000	3173670763357619491332125760000	1241631338584256603066963880000	336905849755015351788257280000	64768133166285425590153440000	8883570925448006755968000000	863421156156879742181760000	58064804687205606850560000	2568230167634490078720000	67176073798769295360000	786891743269632000000	20197470441175049066040900028125

Technical details

Node id: n-a1b8bba2b7403833 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-aed83f03f3b18f2e

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,                 8;
  2:                9,                96,               108;
  3:              225,              2760,              4860,              1920;
  4:            11025,            147840,            325080,            215040,             42000;
  5:           893025,          12768840,          32319000,          27820800,           9450000,           1088640;
  6:        108056025,        1623615840,        4542656580,        4630348800,        2099790000,         431101440,         32598720;
  7:      18261468225,      285628903560,      862690488660,      992500629120,      542972430000,      151472280960,      20765384640,      1107025920;
  8:    4108830350625,    66457879488000,   213385310938800,   269322800947200,   168692263740000,    57537672192000,   10857329683200,   1062744883200,    42032390400;
  9: 1187451971330625, 19765128099717000, 66752382642145200, 90732246046041600, 63059838021180000, 24887560496198400, 5800483885996800, 789287340441600, 57878601580800, 1764322560000;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(64*n^3-224*n^2+240*n-72)*x-(144*n^2-288*n+108)*x^2)*P(n-2,x) + ((64*n^3-224*n^2+240*n-72)*x^2+(64*n^2-128*n+48)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+4+(16*n-8)*x)*P(n-1,x) - (16*n-8)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (64*n^3 - 224*n^2 + 240*n - 72)*x - (144*n^2 - 288*n + 108)*x^2)*P[n - 2, x] + ((64*n^3 - 224*n^2 + 240*n - 72)*x^2 + (64*n^2 - 128*n + 48)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 4 + (16*n - 8)*x)*P[n - 1, x] - (16*n - 8)*x^2*D[P[n - 1, x], 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (64*n^3 - 224*n^2 + 240*n - 72)*x - (144*n^2 - 288*n + 108)*x^2)*v[i - 2] + ((64*n^3 - 224*n^2 + 240*n - 72)*x^2 + (64*n^2 - 128*n + 48)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 4 + (16*n - 8)*x)*v[i - 1] - (16*n - 8)*x^2*deriv(v[i - 1], x)); 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_A013609 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 ?

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.