A143491 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 - 18\,x - 9\,x^{2} - 6\,n - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(7 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 12 + 42\,x + 27\,x^{2}\)

\(P_{1}(x) = 60 + 282\,x + 324\,x^{2} + 108\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	12	42	27								81
1	60	282	324	108							774
2	360	2052	3213	1944	405						7974
3	2520	16524	31725	26460	10125	1458					88812
4	20160	147312	328158	338580	180225	48114	5103				1067652
5	181440	1446768	3616326	4359852	2891700	1081836	214326	17496			13809744
6	1814400	15556320	42673716	58063824	45266445	21228480	5929686	909792	59049		191501712
7	19958400	182005920	539414316	809396928	715670235	396472482	139526226	30338064	3720087	196830	2836699488

All 15 rows

Technical details

Node id: n-f9712f7c5bf7d24d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-38b9e01bfeccb468

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:         12,          42,           27;
  1:         60,         282,          324,          108;
  2:        360,        2052,         3213,         1944,          405;
  3:       2520,       16524,        31725,        26460,        10125,         1458;
  4:      20160,      147312,       328158,       338580,       180225,        48114,        5103;
  5:     181440,     1446768,      3616326,      4359852,      2891700,      1081836,      214326,       17496;
  6:    1814400,    15556320,     42673716,     58063824,     45266445,     21228480,     5929686,      909792,      59049;
  7:   19958400,   182005920,    539414316,    809396928,    715670235,    396472482,   139526226,    30338064,    3720087,    196830;
  8:  239500800,  2303821440,   7291998432,  11870420400,  11623281300,   7334082630,  3061968399,   842432400,  147032010,  14762250,   649539;
  9: 3113510400, 31386683520, 105163176096, 183483458928, 195616733400, 137186886870, 65474878392, 21449798568, 4754625480, 682015950, 57159432, 2125764;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+6*n+9+(6*n+18)*x+9*x^2)*P(n-2,x) + (2*n+7+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 12 + 42*x + 27*x^2, P(1,x) = 60 + 282*x + 324*x^2 + 108*x^3.

%t Mathematica

P[0, x_] := 12 + 42*x + 27*x^2;
P[1, x_] := 60 + 282*x + 324*x^2 + 108*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*P[n - 2, x] + (2*n + 7 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 42*x + 27*x^2; v[2] = 60 + 282*x + 324*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 6*n + 9 + (6*n + 18)*x + 9*x^2)*v[i - 2] + (2*n + 7 + 6*x)*v[i - 1]); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Unsigned 2-Stirling numbers of the first kind.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A136124 → basis A013610 → θ+1

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.