A049029 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	6	1														7
3	60	17	1													78
4	840	315	33	1												1189
5	15120	6825	975	54	1											22975
6	332640	172095	30225	2325	80	1										537366
7	8648640	4979205	1018395	97650	4725	111	1									14748727
8	259459200	163003995	37567845	4240845	258300	8610	147	1								464538943
9	8821612800	5964599025	1515446415	194479110	14056245	594090	14490	188	1							16510802364
10	335221286400	241440974775	66582455625	9489089925	784841400	39602115	1231650	22950	234	1						653559505075
11	14079294028800	10717183201725	3171069022275	493591502250	45591794325	2646140805	98721315	2356200	34650	285	1					28509476802631
12	647647525324800	517786904509875	162928496270925	27357052632525	2773181218500	180544975380	7779649185	223599915	4227300	50325	341	1				1358681712459072
13	32382376266240000	27054779655328425	8990068710597975	1612852285794750	177108838431525	12703154864400	616205329740	20524844340	468783315	7196475	70785	402	1			70230526117482133
14	1748648318376960000	1520395257309303375	530498627448815025	100922649000898725	11885164914823200	926594975431125	49675474648800	1868220834480	49589409870	922161240	11726715	96915	468	1		3913328206245109939
15	101421602465863680000	91451968499625859125	33350312904238204875	6686780216503637250	837802873719937125	70260648366984075	4105825352954325	171109828890000	5141121885900	111374212950	1719577860	18413850	129675	539	1	233823009797734367550

Technical details

Node id: n-6070e7fcac10e9bd · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-fb3bbd519a6eccdb

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:            6,            1;
   3:           60,           17,           1;
   4:          840,          315,          33,          1;
   5:        15120,         6825,         975,         54,         1;
   6:       332640,       172095,       30225,       2325,        80,        1;
   7:      8648640,      4979205,     1018395,      97650,      4725,      111,       1;
   8:    259459200,    163003995,    37567845,    4240845,    258300,     8610,     147,     1;
   9:   8821612800,   5964599025,  1515446415,  194479110,  14056245,   594090,   14490,   188,   1;
  10: 335221286400, 241440974775, 66582455625, 9489089925, 784841400, 39602115, 1231650, 22950, 234, 1;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-2+x)*P(n-1,x) + x*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[(4*n - 2 + x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = (4*n - 2 + x)*v[i - 1] + x*deriv(v[i - 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 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.