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	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

All 15 rows

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.