A290598 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 2\,x + n\right)\,P_{n}(x) = \left(-6 + 4\,x - 2\,x^{2} + 7\,n - 5\,n x + 3\,n^{2}\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 3\,n x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	1														6
3	42	16	1													59
4	490	270	33	1												794
5	7280	5200	936	56	1											13473
6	131040	114400	27040	2400	85	1										274966
7	2766400	2845440	829920	97280	5130	120	1									6544291
8	66946880	79119040	27387360	3979360	277970	9702	161	1								177720474
9	1825824000	2434432000	973772800	168537600	14630000	677600	16800	208	1							5417891009
10	55383328000	82162080000	37246809600	7471833600	774219600	44629200	1470000	27216	261	1						183084397478
11	1848951104000	3018695680000	1528214688000	348311040000	41906172000	2887315200	118482000	2916480	41850	320	1					6789087439851
12	67354647360000	119955419584000	67033910944000	17097718176000	2340759036000	187474980000	9203317200	282429840	5391210	61710	385	1				273979421280346
13	2658263415808000	5126650873344000	3132953311488000	883653498112000	135560479824000	12366920966400	706269379200	25944589440	617728320	9409840	87912	456	1			11950181340737569
14	112976195171840000	234496808465920000	155508409824768000	48038617442816000	8158732582000000	834767165232000	54339501216000	2323482120960	66251362320	1259258000	15661360	121680	533	1		560070261162316854
15	5143739709588480000	11430532687974400000	8172830871901696000	2743327845113856000	510801929493856000	57904672169344000	4230049103280000	206265251512320	6848650929120	155946510720	2420738320	25043200	164346	616	1	28063581037639810643

Technical details

Node id: n-aec89886b0fe4dfc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e38afe273a9e31c4

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:           5,           1;
   3:          42,          16,           1;
   4:         490,         270,          33,          1;
   5:        7280,        5200,         936,         56,         1;
   6:      131040,      114400,       27040,       2400,        85,        1;
   7:     2766400,     2845440,      829920,      97280,      5130,      120,       1;
   8:    66946880,    79119040,    27387360,    3979360,    277970,     9702,     161,     1;
   9:  1825824000,  2434432000,   973772800,  168537600,  14630000,   677600,   16800,   208,   1;
  10: 55383328000, 82162080000, 37246809600, 7471833600, 774219600, 44629200, 1470000, 27216, 261, 1;

%F Formula

The row polynomials satisfy (n+2-2*x)*P(n,x) = (3*n^2+7*n-6-(5*n-4)*x-2*x^2)*P(n-1,x) + (3*n*x-6*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[((3*n^2 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*P[n - 1, x] + (3*n*x - 6*x^2)*D[P[n - 1, x], x])/(n + 2 - 2*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] = ((3*n^2 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*v[i - 1] + (3*n*x - 6*x^2)*deriv(v[i - 1], x))/(n + 2 - 2*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 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.