A088617 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	3	4													7
4	6	20	15												41
5	10	60	105	56											231
6	15	140	420	504	210										1289
7	21	280	1260	2520	2310	792									7183
8	28	504	3150	9240	13860	10296	3003								40081
9	36	840	6930	27720	60060	72072	45045	11440							224143
10	45	1320	13860	72072	210210	360360	360360	194480	43758						1256465
11	55	1980	25740	168168	630630	1441440	2042040	1750320	831402	167960					7059735
12	66	2860	45045	360360	1681680	4900896	9189180	11085360	8314020	3527160	646646				39753273
13	78	4004	75075	720720	4084080	14702688	34918884	55426800	58198140	38798760	14872858	2496144			224298231
14	91	5460	120120	1361360	9189180	39907296	116396280	232792560	320089770	297457160	178474296	62403600	9657700		1267854873
15	105	7280	185640	2450448	19399380	99768240	349188840	853572720	1472412942	1784742960	1487285800	811246800	260757900	37442160	7178461215

Technical details

Node id: n-02c9e638afff8398 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3961bf8dfc4504b6

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 -2)
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:  0;
   2:  1;
   3:  3,    4;
   4:  6,   20,    15;
   5: 10,   60,   105,    56;
   6: 15,  140,   420,   504,    210;
   7: 21,  280,  1260,  2520,   2310,    792;
   8: 28,  504,  3150,  9240,  13860,  10296,   3003;
   9: 36,  840,  6930, 27720,  60060,  72072,  45045,  11440;
  10: 45, 1320, 13860, 72072, 210210, 360360, 360360, 194480, 43758;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + (2*n^2 - 2*n)*x)*P[n - 1, x] + ((2*n - 2)*x + (2*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 + (2*n^2 - 2*n)*x)*v[i - 1] + ((2*n - 2)*x + (2*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n)); 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: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
  - 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.