A281260 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(18\,x^{2} + 27\,n x^{2} - 18\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2 + 6\,x + 8\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	5	24	27													56
3	14	90	180	108												392
4	42	336	945	1080	405											2808
5	132	1260	4536	7560	5670	1458										20616
6	429	4752	20790	45360	51030	27216	5103									154680
7	1430	18018	92664	249480	374220	306180	122472	17496								1181960
8	4862	68640	405405	1297296	2432430	2694384	1683990	524880	59049							9170936
9	16796	262548	1750320	6486480	14594580	20432412	17513496	8660520	2165130	196830						72079112
10	58786	1007760	7482618	31505760	82702620	140107968	153243090	105080976	42220035	8660520	649539					572719672
11	208012	3879876	31744440	149652360	448957080	893188296	1190917728	1050809760	591080490	197026830	33776028	2125764				4593366664
12	742900	14976864	133855722	698377680	2357024670	5387484960	8485288812	9187079616	6698912220	3152429280	886620735	128963016	6908733			37138665208
13	2674440	57946200	561632400	3212537328	12047014980	31112725644	56568592080	72731046960	65457942264	40193473320	16077389328	3868890480	483611310	22320522		302397797256
14	9694845	224652960	2346821100	14602442400	60235074900	173477015712	357796344906	533361011040	572756994810	436386281760	229102797924	78925365792	16442784540	1785641760	71744535	2477524668984

Technical details

Node id: n-f844da1ce197fe2f · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-228b985a796ed6b0

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:     2,      6;
  2:     5,     24,      27;
  3:    14,     90,     180,     108;
  4:    42,    336,     945,    1080,      405;
  5:   132,   1260,    4536,    7560,     5670,     1458;
  6:   429,   4752,   20790,   45360,    51030,    27216,     5103;
  7:  1430,  18018,   92664,  249480,   374220,   306180,   122472,   17496;
  8:  4862,  68640,  405405, 1297296,  2432430,  2694384,  1683990,  524880,   59049;
  9: 16796, 262548, 1750320, 6486480, 14594580, 20432412, 17513496, 8660520, 2165130, 196830;

%F Formula

The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = -(18*n^2-27*n-18)*x^2*P(n-2,x) + (8*n^2-2+(12*n^2+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 27*n - 18)*x^2*P[n - 2, x] + (8*n^2 - 2 + (12*n^2 + 6)*x)*P[n - 1, x])/(2*n^2 + 3*n - 2)]];
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] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 27*n - 18)*x^2*v[i - 2] + (8*n^2 - 2 + (12*n^2 + 6)*x)*v[i - 1])/(2*n^2 + 3*n - 2)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <…?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.