A269944 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 - 8\,x - x^{2} + 32\,n + 8\,n x - 24\,n^{2} - 2\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(5 + 2\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	2	3						5
3	0	8	15	4					27
4	0	72	147	56	5				280
5	0	1152	2460	1092	150	6			4860
6	0	28800	63228	30580	5115	330	7		128060
7	0	1036800	2319408	1185184	222365	18018	637	8	4782420

All 15 rows

Technical details

Node id: n-fd62f258a8ace7f3 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-e5b9891fc1141631

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: 0,          2;
  2: 0,          2,          3;
  3: 0,          8,         15,          4;
  4: 0,         72,        147,         56,         5;
  5: 0,       1152,       2460,       1092,       150,        6;
  6: 0,      28800,      63228,      30580,      5115,      330,       7;
  7: 0,    1036800,    2319408,    1185184,    222365,    18018,     637,      8;
  8: 0,   50803200,  115206192,   61166560,  12377365,  1149720,   52234,   1120,    9;
  9: 0, 3251404800, 7449401088, 4068268096, 868609560, 88434918, 4684316, 131376, 1836, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(2*n^2-8*n+8)*x+x^2)*P(n-2,x) + (2*n^2-6*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*P[n - 2, x] + (2*n^2 - 6*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] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*v[i - 2] + (2*n^2 - 6*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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
  - 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
  - Parent-row provers of polytool polytool ? step › direct check

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.