A204420 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 36\,x - 12\,x^{2} + 108\,n - 120\,n x + 32\,n x^{2} - 144\,n^{2} + 112\,n^{2} x - 16\,n^{2} x^{2} + 80\,n^{3} - 32\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 4\,x - 12\,n + 8\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	2														5
2	33	48	12													93
3	915	1770	900	120												3705
4	47985	109200	76440	20160	1680											255465
5	4085235	10374210	8807400	3250800	529200	30240										27077085
6	513959985	1415383200	1372930020	625363200	143866800	15966720	665280									4088135205
7	89668694115	263116223370	281948366700	148906878120	42756714000	6750263520	544864320	17297280								833709301425
8	20723300222625	64033427858400	74207504170800	43823437257600	14739033909600	2927737612800	337815878400	20756736000	518918400							220813532564625
9	6128733996259875	19775144834294850	24429227538315600	15760339461266400	5978999145319200	1401584457873600	204705525024000	18049019788800	873339667200	17643225600						73697674961035125
10	2257596147560378625	7559487893985627600	9852250038413082300	6833282276332512000	2853162544780548000	759444159560486400	131657238375062400	14763145453056000	1028291296032000	40226554368000	670442572800					30262712632753726125
11	1013604047863441909875	3505368090853418639850	4782682558557153350100	3525089552657241168600	1592537609960365068000	469255252201878542400	92855680782712848000	12416068307119276800	1104045440385888000	62336074312512000	2013339046118400	28158588057600				14994977284195663379625
12	544892544218999884532625	1938895289565656001980400	2752944580094380401529800	2137557378801056309697600	1031949643008887818815600	330675478901081579865600	72773566930870738732800	11154345896253111552000	1187211017134779436800	85845679481802240000	4010881124336486400	108804784254566400	1295295050649600			8822120005095006070085625
13	345558041803937418502921875	1261298608733372499020981250	1854885296504468199326355000	1507052943934553004239250000	770207774760827458465410000	264916877105665305300780000	63665012160683149376160000	10894561597802971408320000	1333886709110819703840000	115813880784844482240000	6948335049011516160000	273210108558266880000	6314563371916800000	64764752532480000		6079936045115691364058578125
14	255286948600337559290901140625	953431818156186611782339800000	1446734069393087770031168932500	1223389149686701404516869760000	657061604629432799591724930000	240194011643266882844522880000	62188492390102898041982760000	11661006560820675423805440000	1599127965590140677051360000	160192829598023847905280000	11571217947960361891200000	585587348858076364800000	19652184126079464960000	391697223316439040000	3497296636753920000	4851718598707800230519186203125

Technical details

Node id: n-6577090cee20b8ba · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-42e9f6f4286ebba9

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:                3,                 2;
  2:               33,                48,                12;
  3:              915,              1770,               900,               120;
  4:            47985,            109200,             76440,             20160,             1680;
  5:          4085235,          10374210,           8807400,           3250800,           529200,            30240;
  6:        513959985,        1415383200,        1372930020,         625363200,        143866800,         15966720,          665280;
  7:      89668694115,      263116223370,      281948366700,      148906878120,      42756714000,       6750263520,       544864320,       17297280;
  8:   20723300222625,    64033427858400,    74207504170800,    43823437257600,   14739033909600,    2927737612800,    337815878400,    20756736000,    518918400;
  9: 6128733996259875, 19775144834294850, 24429227538315600, 15760339461266400, 5978999145319200, 1401584457873600, 204705525024000, 18049019788800, 873339667200, 17643225600;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(32*n^3-112*n^2+120*n-36)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (8*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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (32*n^3 - 112*n^2 + 120*n - 36)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (8*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)

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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.