A122542 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{3}\right)\,P_{n-6}(x) + \left(3\,x^{2} + 3\,x^{3}\right)\,P_{n-5}(x) + \left(3\,x + 3\,x^{2} + 3\,x^{3}\right)\,P_{n-4}(x) + \left(1 - 3\,x - 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 3\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1 + 2\,x\)

\(P_{3}(x) = 2 + 8\,x + 3\,x^{2}\)

\(P_{4}(x) = 2 + 20\,x + 21\,x^{2} + 4\,x^{3}\)

\(P_{5}(x) = 2 + 36\,x + 78\,x^{2} + 40\,x^{3} + 5\,x^{4}\)

\(P_{6}(x) = 2 + 52\,x + 198\,x^{2} + 200\,x^{3} + 65\,x^{4} + 6\,x^{5}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	2	8	3													13
4	2	20	21	4												47
5	2	36	78	40	5											161
6	2	52	198	200	65	6										523
7	2	68	390	664	410	96	7									1637
8	2	84	654	1672	1690	732	133	8								4975
9	2	100	990	3480	5210	3612	1190	176	9							14769
10	2	116	1398	6344	13050	13212	6846	1808	225	10						43011
11	2	132	1878	10520	28090	38844	29022	11888	2610	280	11					123277
12	2	148	2430	16264	54010	96924	98182	57360	19314	3620	341	12				348607
13	2	164	3054	23832	95290	213276	279510	219760	104562	29780	4862	408	13			974513
14	2	180	3750	33480	157210	425436	695310	704272	447858	178820	44022	6360	481	14		2697195
15	2	196	4518	45464	245850	784956	1553678	1959024	1597554	847220	290422	62856	8138	560	15	7400453

Technical details

Node id: n-62c8c2cd2df5630e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7b9a7ea9d38360c0

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: 1,   2;
   3: 2,   8,    3;
   4: 2,  20,   21,    4;
   5: 2,  36,   78,   40,     5;
   6: 2,  52,  198,  200,    65,     6;
   7: 2,  68,  390,  664,   410,    96,    7;
   8: 2,  84,  654, 1672,  1690,   732,  133,    8;
   9: 2, 100,  990, 3480,  5210,  3612, 1190,  176,   9;
  10: 2, 116, 1398, 6344, 13050, 13212, 6846, 1808, 225, 10;

%F Formula

The row polynomials satisfy P(n,x) = x^3*P(n-6,x) + (3*x^2+3*x^3)*P(n-5,x) + (3*x+3*x^2+3*x^3)*P(n-4,x) + (1-3*x-3*x^2+x^3)*P(n-3,x) - (3+3*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 2 + 8*x + 3*x^2, P(4,x) = 2 + 20*x + 21*x^2 + 4*x^3, P(5,x) = 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4, P(6,x) = 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 2 + 8*x + 3*x^2;
P[4, x_] := 2 + 20*x + 21*x^2 + 4*x^3;
P[5, x_] := 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4;
P[6, x_] := 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5;
P[n_, x_] := P[n, x] = Expand[x^3*P[n - 6, x] + (3*x^2 + 3*x^3)*P[n - 5, x] + (3*x + 3*x^2 + 3*x^3)*P[n - 4, x] + (1 - 3*x - 3*x^2 + x^3)*P[n - 3, x] - (3 + 3*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, 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] = 1 + 2*x; v[3] = 2 + 8*x + 3*x^2; v[4] = 2 + 20*x + 21*x^2 + 4*x^3; v[5] = 2 + 36*x + 78*x^2 + 40*x^3 + 5*x^4; v[6] = 2 + 52*x + 198*x^2 + 200*x^3 + 65*x^4 + 6*x^5; for(n = 7, N, my(i = n); v[i] = x^3*v[i - 6] + (3*x^2 + 3*x^3)*v[i - 5] + (3*x + 3*x^2 + 3*x^3)*v[i - 4] + (1 - 3*x - 3*x^2 + x^3)*v[i - 3] - (3 + 3*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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 one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0,…?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.