A102365 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	3	1													4
3	15	9	1												25
4	105	90	21	1											217
5	945	1050	375	45	1										2416
6	10395	14175	6825	1350	93	1									32839
7	135135	218295	132300	36750	4515	189	1								527185
8	2027025	3783780	2765070	992250	178605	14490	381	1							9761602
9	34459425	72972900	62432370	27505170	6568695	815850	45375	765	1						204800551
10	654729075	1550674125	1520268750	794593800	237286665	40186125	3581025	139950	1533	1					4801461049
11	13749310575	36010099125	39800635875	24081057000	8646883245	1865767365	233159850	15303750	427515	3069	1				124402647370
12	316234143225	907454497950	1116313072875	767583691875	322351704675	84783158460	13759362540	1303533000	64207605	1297890	6141	1			3529848676237
13	7905853580625	24666263171550	33424574007825	25747220874375	12388232206350	3850561014300	772353822240	96895038240	7095533445	265915650	3924375	12285	1		108859319101261
14	213458046676875	719432675836875	1064760360238575	908361952447950	492848620664400	176953627100250	42389795448000	6666112252800	659421097335	37870908075	1091133225	11834550	24573	1	3625569585663484

Technical details

Node id: n-638f95f654523b99 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d54b1929db83100a

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:        1;
  1:        1;
  2:        3,        1;
  3:       15,        9,        1;
  4:      105,       90,       21,        1;
  5:      945,     1050,      375,       45,       1;
  6:    10395,    14175,     6825,     1350,      93,      1;
  7:   135135,   218295,   132300,    36750,    4515,    189,     1;
  8:  2027025,  3783780,  2765070,   992250,  178605,  14490,   381,   1;
  9: 34459425, 72972900, 62432370, 27505170, 6568695, 815850, 45375, 765, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (2*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (n - 1)*x)*P[n - 1, x] - (2*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + (n - 1)*x)*v[i - 1] - (2*x + x^2)*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 - 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …?
    
    ↑ Liu–Wang theorem (above)

References

[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