A370258 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 10\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(20 + 60\,x + 18\,n + 66\,n x + 4\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(14\,x + 15\,x^{2} + 8\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 10 + 30\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	10	30															40
1	21	168	252														441
2	36	540	1980	1980													4536
3	55	1320	8580	20020	15015												44990
4	78	2730	27300	109200	185640	111384											436332
5	105	5040	71400	428400	1220940	1627920	813960										4167765
6	136	8568	162792	1356600	5697720	12534984	13728792	5883768									39373360
7	171	13680	335160	3686760	21198870	67836384	121136400	112483800	42181425								368872650
8	210	20790	637560	8925840	66943800	290089800	745945200	1118917800	901350450	300450150							3433281600
9	253	30360	1138500	19734000	186486300	1044323280	3605401800	7725861000	9979237125	7096346400	2128903920						31787462938
10	300	42900	1930500	40540500	470269800	3291888600	14578363800	41652468000	76362858000	86544572400	55073818800	15020132400					293036886000
11	351	58968	3135132	78378300	1093377285	9330152832	51315840576	186936276384	454359005100	726974408160	733583266416	422366123088	105591530772				2691631553364
12	406	79170	4908540	143983840	2375733360	24232480272	161549868480	726974408160	2241504425160	4732064897560	6710928400176	6100844000160	3206853897520	740043207120			24647520289924
13	465	104160	7447440	253212960	4874349480	58492193760	463759536240	2517551768160	9545717120940	25455245655840	47439321449520	60377318208480	49927397749320	24140719790880	5173011383760		225103669971405
14	528	134640	10995600	428828400	9519990480	132645200688	1231705434960	7918106367600	36071373452400	117833153277840	276372304960752	460620508267920	531485201847600	402994273928400	180387913091760	36077582618352	2051134728397920

Technical details

Node id: n-0e4c3979a5ed69fd · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-23ffd7df897ade57

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:  10,    30;
  1:  21,   168,     252;
  2:  36,   540,    1980,     1980;
  3:  55,  1320,    8580,    20020,     15015;
  4:  78,  2730,   27300,   109200,    185640,     111384;
  5: 105,  5040,   71400,   428400,   1220940,    1627920,     813960;
  6: 136,  8568,  162792,  1356600,   5697720,   12534984,   13728792,    5883768;
  7: 171, 13680,  335160,  3686760,  21198870,   67836384,  121136400,  112483800,   42181425;
  8: 210, 20790,  637560,  8925840,  66943800,  290089800,  745945200, 1118917800,  901350450,  300450150;
  9: 253, 30360, 1138500, 19734000, 186486300, 1044323280, 3605401800, 7725861000, 9979237125, 7096346400, 2128903920;

%F Formula

The row polynomials satisfy (4*n^2+10*n+6)*P(n,x) = (4*n^2+18*n+20+(18*n^2+66*n+60)*x)*P(n-1,x) + ((8*n+14)*x+(9*n+15)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 30*x.

%t Mathematica

P[0, x_] := 10 + 30*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 18*n + 20 + (18*n^2 + 66*n + 60)*x)*P[n - 1, x] + ((8*n + 14)*x + (9*n + 15)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 10*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 10 + 30*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 18*n + 20 + (18*n^2 + 66*n + 60)*x)*v[i - 1] + ((8*n + 14)*x + (9*n + 15)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 10*n + 6)); 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, k)*binomial(2*n+k, k), 0 <= k <= n.?
  - 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

Related ?

Same fingerprint ?

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.