A154602 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	7	6	1						14
4	30	37	12	1					80
5	153	244	118	20	1				536
6	898	1741	1120	290	30	1			4080
7	5919	13442	10771	3780	605	42	1		34560
8	43102	111833	106988	47551	10430	1127	56	1	321088

All 15 rows

Technical details

Node id: n-581a1348a1a48f0a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d52f91d92fc1b696

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:       2,       1;
   3:       7,       6,        1;
   4:      30,      37,       12,       1;
   5:     153,     244,      118,      20,       1;
   6:     898,    1741,     1120,     290,      30,      1;
   7:    5919,   13442,    10771,    3780,     605,     42,     1;
   8:   43102,  111833,   106988,   47551,   10430,   1127,    56,    1;
   9:  342673,  997416,  1106540,  595896,  167286,  24920,  1932,   72,  1;
  10: 2946946, 9489145, 11953248, 7560820, 2598036, 498414, 53424, 3108, 90, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-2,x) + (2+4*x+2*x^2)*d/dx P(n-2,x) + 2*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x] + (2 + 4*x + 2*x^2)*D[P[n - 2, x], x] + 2*P[n - 1, x] + (2 + 2*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; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (1 + 2*x + x^2)*v[i - 2] + (2 + 4*x + 2*x^2)*deriv(v[i - 2], x) + 2*v[i - 1] + (2 + 2*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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].?
  - Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
  - 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.