A290596 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	20	12	1													33
4	200	168	27	1												396
5	2640	2816	660	48	1											6165
6	43120	55440	16940	1820	75	1										117396
7	837760	1256640	471240	66640	4080	108	1									2636469
8	18849600	32253760	14294280	2499000	202300	7980	147	1								68107068
9	481712000	924887040	472077760	98094080	9853200	515200	14168	192	1							1987153641
10	13776963200	29307358080	16908091200	4064773440	486748080	31646160	1155336	23400	243	1						64576759140
11	435853017600	1016990374400	653779526400	178303507200	24764376000	1922880960	87403680	2352480	36540	300	1					2311703475561
12	15109571276800	38355065548800	27168171430400	8281207334400	1307559052800	117936698880	6409603200	214980480	4440480	54560	363	1				90346140421164
13	569514609664000	1562097215078400	1208184564787200	406796149760000	71915747904000	7383862886400	466619112960	18727188480	482328000	7884800	78540	432	1			3826377986673213
14	23187380536320000	68325481313689600	57272829924710400	21100516288051200	4126801654976000	474951597120000	34196514992640	1596938703360	49242473280	1004203200	13313300	109668	507	1		174523805028663156
15	1014061442121728000	3194293542683443200	2883737226033664000	1153494890413465600	247177476517171200	31503011712972800	2544474022970880	135464455526400	4862826608640	118775937280	1965043080	21550256	149240	588	1	8526952511550231165

Technical details

Node id: n-6d54ebf30881a6d6 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3e125a4b64c05092

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:           3,           1;
   3:          20,          12,           1;
   4:         200,         168,          27,          1;
   5:        2640,        2816,         660,         48,         1;
   6:       43120,       55440,       16940,       1820,        75,        1;
   7:      837760,     1256640,      471240,      66640,      4080,      108,       1;
   8:    18849600,    32253760,    14294280,    2499000,    202300,     7980,     147,     1;
   9:   481712000,   924887040,   472077760,   98094080,   9853200,   515200,   14168,   192,   1;
  10: 13776963200, 29307358080, 16908091200, 4064773440, 486748080, 31646160, 1155336, 23400, 243, 1;

%F Formula

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