A102662 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9\,x^{2}\right)\,P_{n-4}(x) + \left(4\,x^{3}\right)\,P_{n-4}'(x) + \left(-4\,x + 18\,x^{2}\right)\,P_{n-3}(x) + \left(4\,x^{2} + 8\,x^{3}\right)\,P_{n-3}'(x) + \left(-1 + 9\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3}\right)\,P_{n-2}'(x) + \left(2 + 4\,x\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 3 + 20\,x + 9\,x^{2}\)

\(P_{3}(x) = 3 + 44\,x + 63\,x^{2} + 16\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	4														7
2	3	20	9													32
3	3	44	63	16												126
4	3	68	207	144	25											447
5	3	92	459	624	275	36										1489
6	3	116	819	1808	1475	468	49									4738
7	3	140	1287	4080	5275	2988	735	64								14572
8	3	164	1863	7824	14475	12708	5439	1088	81							43645
9	3	188	2547	13424	33075	41148	26803	9152	1539	100						127979
10	3	212	3339	21264	66275	109620	100107	51264	14499	2100	121					368804
11	3	236	4239	31728	120475	252684	305319	217024	90963	21900	2783	144				1047498
12	3	260	5247	45200	203275	521604	798455	746560	430515	152100	31823	3600	169			2938811
13	3	284	6363	62064	323475	987804	1852347	2188224	1650051	795900	242363	44784	4563	196		8158421
14	3	308	7587	82704	491075	1746324	3906819	5650496	5364387	3364500	1389443	371088	61347	5684	225	22441990

Technical details

Node id: n-7e6f7fcdae976fd5 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f42ae7bdcb1fb143

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 0)
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: 3,   4;
  2: 3,  20,    9;
  3: 3,  44,   63,    16;
  4: 3,  68,  207,   144,    25;
  5: 3,  92,  459,   624,   275,    36;
  6: 3, 116,  819,  1808,  1475,   468,    49;
  7: 3, 140, 1287,  4080,  5275,  2988,   735,   64;
  8: 3, 164, 1863,  7824, 14475, 12708,  5439, 1088,   81;
  9: 3, 188, 2547, 13424, 33075, 41148, 26803, 9152, 1539, 100;

%F Formula

The row polynomials satisfy P(n,x) = 9*x^2*P(n-4,x) + 4*x^3*d/dx P(n-4,x) - (4*x-18*x^2)*P(n-3,x) + (4*x^2+8*x^3)*d/dx P(n-3,x) - (1-9*x^2)*P(n-2,x) + 4*x^3*d/dx P(n-2,x) + (2+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 3 + 20*x + 9*x^2, P(3,x) = 3 + 44*x + 63*x^2 + 16*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 3 + 20*x + 9*x^2;
P[3, x_] := 3 + 44*x + 63*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[9*x^2*P[n - 4, x] + 4*x^3*D[P[n - 4, x], x] - (4*x - 18*x^2)*P[n - 3, x] + (4*x^2 + 8*x^3)*D[P[n - 3, x], x] - (1 - 9*x^2)*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] + (2 + 4*x)*P[n - 1, x] - 4*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; v[2] = 3 + 4*x; v[3] = 3 + 20*x + 9*x^2; v[4] = 3 + 44*x + 63*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = 9*x^2*v[i - 4] + 4*x^3*deriv(v[i - 4], x) - (4*x - 18*x^2)*v[i - 3] + (4*x^2 + 8*x^3)*deriv(v[i - 3], x) - (1 - 9*x^2)*v[i - 2] + 4*x^3*deriv(v[i - 2], x) + (2 + 4*x)*v[i - 1] - 4*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 (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[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.