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	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	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
8	9	630	10395	69300	225225	378378	315315	102960							1102212
9	10	880	18480	160160	700700	1681680	2242240	1555840	437580						6797570
10	11	1188	30888	336336	1891890	6054048	11435424	12602304	7482618	1847560					41682267
11	12	1560	49140	655200	4586400	18712512	46781280	72558720	68023800	35271600	7759752				254399976
12	13	2002	75075	1201200	10210200	51459408	162954792	332560800	436486050	355655300	163601438	32449872			1546656150
13	14	2520	110880	2094400	21205800	128931264	501399360	1289312640	2216006100	2516945200	1812200544	748843200	135207800		9372259722
14	15	3120	159120	3500640	41570100	299304720	1396755360	4389802560	9465511770	14022980400	14022980400	9039607200	3389852700	561632400	56633660505

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.