A281260 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(8\,x^{2} + 12\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 4\,x\)

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	4														6
2	5	16	12													33
3	14	60	80	32												186
4	42	224	420	320	80											1086
5	132	840	2016	2240	1120	192										6540
6	429	3168	9240	13440	10080	3584	448									40389
7	1430	12012	41184	73920	73920	40320	10752	1024								254562
8	4862	45760	180180	384384	480480	354816	147840	30720	2304							1631346
9	16796	175032	777920	1921920	2882880	2690688	1537536	506880	84480	5120						10599252
10	58786	671840	3325608	9335040	16336320	18450432	13453440	6150144	1647360	225280	11264					69665514
11	208012	2586584	14108640	44341440	88682880	117621504	104552448	61501440	23063040	5125120	585728	24576				462401412
12	742900	9984576	59491432	206926720	465585120	709463040	744936192	537698304	261381120	82001920	15375360	1490944	53248			3095130876
13	2674440	38630800	249614400	951862912	2379657280	4097149056	4966241280	4256778240	2554066944	1045524480	278806528	44728320	3727360	114688		20869576728
14	9694845	149768640	1043031600	4326649600	11898286400	22844709888	31411476096	31216373760	22348085760	11351408640	3972993024	912457728	126730240	9175040	245760	141621087021

Technical details

Node id: n-e14fba9d75956da0 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-c980e8256bc08462

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,      4;
  2:     5,     16,     12;
  3:    14,     60,     80,      32;
  4:    42,    224,    420,     320,      80;
  5:   132,    840,   2016,    2240,    1120,     192;
  6:   429,   3168,   9240,   13440,   10080,    3584,     448;
  7:  1430,  12012,  41184,   73920,   73920,   40320,   10752,   1024;
  8:  4862,  45760, 180180,  384384,  480480,  354816,  147840,  30720,  2304;
  9: 16796, 175032, 777920, 1921920, 2882880, 2690688, 1537536, 506880, 84480, 5120;

%F Formula

The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = -(8*n^2-12*n-8)*x^2*P(n-2,x) + (8*n^2-2+(8*n^2+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 12*n - 8)*x^2*P[n - 2, x] + (8*n^2 - 2 + (8*n^2 + 4)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 - 12*n - 8)*x^2*v[i - 2] + (8*n^2 - 2 + (8*n^2 + 4)*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.