A159854 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + x + n + n x\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} + 2\,n + 3\,n x + n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	2								4
1	5	8	3							16
2	12	26	18	4						60
3	28	76	75	32	5					216
4	64	208	264	164	50	6				756
5	144	544	840	680	305	72	7			2592
6	320	1376	2496	2480	1460	510	98	8		8748
7	704	3392	7056	8288	6020	2772	791	128	9	29160

All 15 rows

Technical details

Node id: n-e96218acbdd6a978 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ddfce4c1f50f5030

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    2,     2;
  1:    5,     8,     3;
  2:   12,    26,    18,     4;
  3:   28,    76,    75,    32,     5;
  4:   64,   208,   264,   164,    50,     6;
  5:  144,   544,   840,   680,   305,    72,     7;
  6:  320,  1376,  2496,  2480,  1460,   510,    98,    8;
  7:  704,  3392,  7056,  8288,  6020,  2772,   791,  128,    9;
  8: 1536,  8192, 19200, 25984, 22400, 12768,  4816, 1160,  162,  10;
  9: 3328, 19456, 50688, 77568, 77280, 52416, 24528, 7824, 1629, 200, 11;

%F Formula

The row polynomials satisfy (n+3+(n+1)*x)*P(n,x) = (2*n+8+(3*n+8)*x+(n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (3*n + 8)*x + (n + 2)*x^2)*P[n - 1, x])/(n + 3 + (n + 1)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (3*n + 8)*x + (n + 2)*x^2)*v[i - 1])/(n + 3 + (n + 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 (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of Riordan array (1-x,x/(1-x)).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Row-hypergeometric rows (multiplier sequences) (above)

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.