A166960 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + x^{3} - 2\,n x^{2} + n x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - n x^{3}\right)\,P_{n-2}''(x) + \left(1 + 2\,n x\right)\,P_{n-1}(x) + \left(3\,x - x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	12	3													16
3	1	54	63	4												122
4	1	224	810	232	5											1272
5	1	906	8634	6952	705	6										17204
6	1	3636	84501	157280	43695	1908	7									291028
7	1	14558	793233	3079820	1878775	225990	4795	8								5997180
8	1	58248	7291956	55622984	66218250	17153760	1025920	11472	9							147382600
9	1	233010	66326580	958026000	2072628630	1014844860	130320960	4251648	26541	10						4246658240
10	1	932060	600084855	16035899520	60198443250	51457341096	12305655390	867858432	16499133	59980	11					141482773728
11	1	3728262	5414744595	263775410580	1665320076450	2358131202756	963178501782	125860970448	5241792717	60995110	133287	12				5386987556000
12	1	14913072	48794217678	4292603163880	44600475280275	100879796033136	66453818076492	14659754692320	1132553168847	29396367520	217411590	292536	13			232097423617360
13	1	59652314	439416394398	69397299148024	1168669421555375	4113357790219866	4197781848724044	1469854844920416	190690150849767	9231598801150	155650819478	753654024	636025	14		11219578835374896
14	1	238609284	3955910769705	1117387448678752	30170948402169705	162104913506579676	248881567384786749	132450429834669312	27021009122579403	2194427552446780	69607542563163	787729755360	2556863803	1372644	15	604015037231844352

Technical details

Node id: n-0aa890aa63c05998 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-813e3a68dec52ec0

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: 1,      2;
  2: 1,     12,        3;
  3: 1,     54,       63,         4;
  4: 1,    224,      810,       232,          5;
  5: 1,    906,     8634,      6952,        705,          6;
  6: 1,   3636,    84501,    157280,      43695,       1908,         7;
  7: 1,  14558,   793233,   3079820,    1878775,     225990,      4795,       8;
  8: 1,  58248,  7291956,  55622984,   66218250,   17153760,   1025920,   11472,     9;
  9: 1, 233010, 66326580, 958026000, 2072628630, 1014844860, 130320960, 4251648, 26541, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-2,x) - ((2*n+2)*x^2-(n+1)*x^3)*d/dx P(n-2,x) - (n+1)*x^3*d^2/dx^2 P(n-2,x) + (1+2*n*x)*P(n-1,x) + (3*x-x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 2, x] - ((2*n + 2)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] - (n + 1)*x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*n*x)*P[n - 1, x] + (3*x - x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 2] - ((2*n + 2)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) - (n + 1)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*n*x)*v[i - 1] + (3*x - x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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 T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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.