A281260 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(32\,x^{2} + 48\,n x^{2} - 32\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2 + 8\,x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	8														10
2	5	32	48													85
3	14	120	320	256												710
4	42	448	1680	2560	1280											6010
5	132	1680	8064	17920	17920	6144										51860
6	429	6336	36960	107520	161280	114688	28672									455885
7	1430	24024	164736	591360	1182720	1290240	688128	131072								4073710
8	4862	91520	720720	3075072	7687680	11354112	9461760	3932160	589824							36917710
9	16796	350064	3111680	15375360	46126080	86102016	98402304	64880640	21626880	2621440						338613260
10	58786	1343680	13302432	74680320	261381120	590413824	861020160	787218432	421724160	115343360	11534336					3138020610
11	208012	5173168	56434560	354731520	1418926080	3763888128	6691356672	7872184320	5904138240	2624061440	599785472	50331648				29341219260
12	742900	19969152	237965728	1655413760	7449361920	22702817280	47675916288	68825382912	66913566720	41984983040	15744368640	3053453312	218103808			276482045460
13	2674440	77261600	998457600	7614903296	38074516480	131108769792	317839441920	544867614720	653841137664	535308533760	285497884672	91603599360	15267266560	939524096		2623041585960
14	9694845	299537280	4172126400	34613196800	190372582400	731030716416	2010334470144	3995695841280	5721109954560	5811921223680	4068344856576	1868713426944	519087063040	75161927680	4026531840	25034893149885

Technical details

Node id: n-03750a627f7ed2dc · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-80fa4861659be6fe

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,      8;
  2:     5,     32,      48;
  3:    14,    120,     320,      256;
  4:    42,    448,    1680,     2560,     1280;
  5:   132,   1680,    8064,    17920,    17920,     6144;
  6:   429,   6336,   36960,   107520,   161280,   114688,    28672;
  7:  1430,  24024,  164736,   591360,  1182720,  1290240,   688128,   131072;
  8:  4862,  91520,  720720,  3075072,  7687680, 11354112,  9461760,  3932160,   589824;
  9: 16796, 350064, 3111680, 15375360, 46126080, 86102016, 98402304, 64880640, 21626880, 2621440;

%F Formula

The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = -(32*n^2-48*n-32)*x^2*P(n-2,x) + (8*n^2-2+(16*n^2+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(32*n^2 - 48*n - 32)*x^2*P[n - 2, x] + (8*n^2 - 2 + (16*n^2 + 8)*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(32*n^2 - 48*n - 32)*x^2*v[i - 2] + (8*n^2 - 2 + (16*n^2 + 8)*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.