A286724 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	1							4
3	16	10	1						27
4	120	108	21	1					250
5	1152	1344	384	36	1				2917
6	13440	19200	7200	1000	55	1			40896
7	184320	311040	144000	26400	2160	78	1		667999
8	2903040	5644800	3104640	705600	76440	4116	105	1	12438742

All 15 rows

Technical details

Node id: n-2e7006cec065c803 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-428eb0e1ac93bf64

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          1;
   2:          3,          1;
   3:         16,         10,          1;
   4:        120,        108,         21,         1;
   5:       1152,       1344,        384,        36,        1;
   6:      13440,      19200,       7200,      1000,       55,       1;
   7:     184320,     311040,     144000,     26400,     2160,      78,      1;
   8:    2903040,    5644800,    3104640,    705600,    76440,    4116,    105,     1;
   9:   51609600,  113541120,   72253440,  19568640,  2634240,  188160,   7168,   136,   1;
  10: 1021870080, 2508226560, 1811496960, 568995840, 91445760, 8128512, 411264, 11664, 171, 1;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 2 - (n - 1)*x - x^2)*P[n - 1, x] + ((2*n - 2)*x - 2*x^2)*D[P[n - 1, x], x])/(n - x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 2 - (n - 1)*x - x^2)*v[i - 1] + ((2*n - 2)*x - 2*x^2)*deriv(v[i - 1], x))/(n - x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows. A generalization of unsigned Lah numbers, called L[2,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - 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

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.