A386789 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	3	12						15
3	4	40	60					104
4	5	90	315	280				690
5	6	168	1008	2016	1260			4458
6	7	280	2520	8400	11550	5544		28301
7	8	432	5400	26400	59400	61776	24024	177440

All 15 rows

Technical details

Node id: n-9d7951b6c7c263fc · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6832a290ef8b97c2

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  2;
  2:  3,  12;
  3:  4,  40,    60;
  4:  5,  90,   315,    280;
  5:  6, 168,  1008,   2016,   1260;
  6:  7, 280,  2520,   8400,  11550,    5544;
  7:  8, 432,  5400,  26400,  59400,   61776,   24024;
  8:  9, 630, 10395,  69300, 225225,  378378,  315315,  102960;
  9: 10, 880, 18480, 160160, 700700, 1681680, 2242240, 1555840, 437580;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 1 + (2*n^2 + 2*n)*x)*P[n - 1, x] + (2*n*x + 2*n*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 1 + (2*n^2 + 2*n)*x)*v[i - 1] + (2*n*x + 2*n*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] 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

Related ?

Same fingerprint ?

References

[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.