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	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

All 15 rows

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.