A290598 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - x + n x\right)\,P_{n}(x) = \left(4 - 2\,x - 6\,n - 5\,n x + n x^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 3\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	2														6
3	28	28	3													59
4	280	420	90	4												794
5	3640	7280	2340	208	5											13473
6	58240	145600	62400	8320	400	6										274966
7	1106560	3319680	1778400	316160	22800	684	7									6544291
8	24344320	85205120	54774720	12172160	1170400	52668	1078	8								177720474
9	608608000	2434432000	1825824000	486886400	58520000	3511200	107800	1600	9							5417891009
10	17041024000	76684608000	65729664000	20449228800	2949408000	221205600	9055200	201600	2268	10						183084397478
11	528271744000	2641358720000	2547024480000	905608704000	152386080000	13714747200	701778000	20832000	351540	3100	11					6789087439851
12	17961239296000	98786816128000	105843017280000	42337206912000	8141770560000	854885908800	52492994400	1947792000	43825320	579700	4114	12				273979421280346
13	664565853952000	3987395123712000	4699429967232000	2088635540992000	451868266080000	54224191929600	3884481585600	172963929600	4864610520	85795600	913308	5328	13			11950181340737569
14	26582634158080000	172787122027520000	222154871178240000	108609048131584000	26107944262400000	3524572475424000	288561489216000	14990207232000	505919494080	11153428000	158306720	1385280	6760	14		560070261162316854
15	1143053268797440000	8001372881582080000	11144769370775040000	5943876997746688000	1571698244596480000	235754736689472000	21714252063504000	1289157821952000	50760589239360	1342872731200	23825161360	277979520	2034760	8428	15	28063581037639810643

Technical details

Node id: n-ed7f37079a43bdf8 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9c7ee164876013bf

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:           1;
   2:           4,           2;
   3:          28,          28,           3;
   4:         280,         420,          90,           4;
   5:        3640,        7280,        2340,         208,          5;
   6:       58240,      145600,       62400,        8320,        400,         6;
   7:     1106560,     3319680,     1778400,      316160,      22800,       684,       7;
   8:    24344320,    85205120,    54774720,    12172160,    1170400,     52668,    1078,      8;
   9:   608608000,  2434432000,  1825824000,   486886400,   58520000,   3511200,  107800,   1600,    9;
  10: 17041024000, 76684608000, 65729664000, 20449228800, 2949408000, 221205600, 9055200, 201600, 2268, 10;

%F Formula

The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(6*n-4-(3*n^2-5*n-2)*x-n*x^2)*P(n-1,x) - (6*x-3*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 4 - (3*n^2 - 5*n - 2)*x - n*x^2)*P[n - 1, x] - (6*x - 3*n*x^2)*D[P[n - 1, x], x])/(-2 + (n - 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(6*n - 4 - (3*n^2 - 5*n - 2)*x - n*x^2)*v[i - 1] - (6*x - 3*n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 1)*x)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.