A143491 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 4\,x - x^{2} - 4\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x + 2\,n\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 6 + 10\,x + 3\,x^{2}\)

\(P_{1}(x) = 24 + 52\,x + 27\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	6	10	3								19
1	24	52	27	4							107
2	120	308	213	56	5						702
3	720	2088	1740	620	100	6					5274
4	5040	16056	15312	6660	1475	162	7				44712
5	40320	138528	146580	73696	20125	3066	245	8			422568
6	362880	1327392	1527012	858704	273245	51744	5782	352	9		4407120
7	3628800	13999680	17261208	10623056	3805830	845334	118188	10128	486	10	50292720

All 15 rows

Technical details

Node id: n-c71dcd281e71f7f9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-13dd874d79e2c744

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:         6,         10,          3;
  1:        24,         52,         27,          4;
  2:       120,        308,        213,         56,         5;
  3:       720,       2088,       1740,        620,       100,         6;
  4:      5040,      16056,      15312,       6660,      1475,       162,        7;
  5:     40320,     138528,     146580,      73696,     20125,      3066,      245,       8;
  6:    362880,    1327392,    1527012,     858704,    273245,     51744,     5782,     352,      9;
  7:   3628800,   13999680,   17261208,   10623056,   3805830,    845334,   118188,   10128,    486,    10;
  8:  39916800,  161254080,  210872808,  139868560,  55142950,  13865670,  2286291,  246480,  16740,   650,  11;
  9: 479001600, 2014882560, 2772354816, 1959586464, 836551100, 232559580, 43612107, 5570664, 478170, 26400, 847, 12;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 6 + 10*x + 3*x^2, P(1,x) = 24 + 52*x + 27*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 6 + 10*x + 3*x^2;
P[1, x_] := 24 + 52*x + 27*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*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] = 6 + 10*x + 3*x^2; v[2] = 24 + 52*x + 27*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*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 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

Related ?

Same fingerprint ?

A136124 → 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.