A290598 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - x - 5\,n + n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(-3\,x + 3\,n x\right)\,P_{n-1}'(x)\)

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

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

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	14	2													16
4	210	60	3												273
5	3640	1560	156	4											5360
6	72800	41600	6240	320	5										120965
7	1659840	1185600	237120	18240	570	6									3101376
8	42602560	36516480	9129120	936320	43890	924	7								89229301
9	1217216000	1217216000	365164800	46816000	2926000	92400	1400	8							2849432608
10	38342304000	43819776000	15336921600	2359526400	184338000	7761600	176400	2016	9						100050806025
11	1320679360000	1698016320000	679206528000	121908864000	11428956000	601524000	18228000	312480	2790	10					3831860095280
12	49393408064000	70562011520000	31752905184000	6513416448000	712404924000	44993995200	1704318000	38955840	521730	3740	11				158980883934521
13	1993697561856000	3132953311488000	1566476655744000	361494612864000	45186826608000	3329555644800	151343438400	4324098240	77216040	830280	4884	12			7103294269792656
14	86393561013760000	148103247452160000	81456786098688000	20886355409920000	2937143729520000	247338419328000	13116431328000	449706216960	10038085200	143915200	1269840	6240	13		340038008444197453
15	4000686440791040000	7429846247183360000	4457907748310016000	1257358595677184000	196462280574560000	18612216054432000	1128013094208000	45120523768320	1208585458080	21659237600	254814560	1878240	7826	14	17362047892709964640

Technical details

Node id: n-931e17ea71b7d537 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0d3b2fa309e19f30

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:          14,           2;
   4:         210,          60,           3;
   5:        3640,        1560,         156,          4;
   6:       72800,       41600,        6240,        320,         5;
   7:     1659840,     1185600,      237120,      18240,       570,       6;
   8:    42602560,    36516480,     9129120,     936320,     43890,     924,      7;
   9:  1217216000,  1217216000,   365164800,   46816000,   2926000,   92400,   1400,    8;
  10: 38342304000, 43819776000, 15336921600, 2359526400, 184338000, 7761600, 176400, 2016, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (3*n^2-5*n+2+(n-1)*x)*P(n-1,x) + (3*n-3)*x*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[((3*n^2 - 5*n + 2 + (n - 1)*x)*P[n - 1, x] + (3*n - 3)*x*D[P[n - 1, x], x])/(n - 2)]];
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] = ((3*n^2 - 5*n + 2 + (n - 1)*x)*v[i - 1] + (3*n - 3)*x*deriv(v[i - 1], x))/(n - 2)); 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[3,2].?
  - 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.