A330381 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(9 + 3\,x - 9\,x^{2}\right)\,P_{n-3}(x) + \left(1 - 3\,x - 3\,x^{2} - 4\,x^{3}\right)\,P_{n-3}'(x) + \left(-22 - 2\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(-2 + 6\,x + 8\,x^{2}\right)\,P_{n-2}'(x) + \left(15 + 8\,x\right)\,P_{n-1}(x) + \left(1 - 4\,x\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

\(P_{2}(x) = 6 + 11\,x + 3\,x^{2}\)

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	2														5
2	6	11	3													20
3	10	35	25	4												74
4	15	84	113	46	5											263
5	21	170	363	276	75	6										911
6	28	307	941	1131	571	113	7									3098
7	36	511	2107	3623	2885	1057	161	8								10388
8	45	800	4242	9780	11123	6420	1806	220	9							34445
9	55	1194	7878	23280	35441	29194	12930	2904	291	10						113177
10	66	1715	13731	50322	98020	107993	68245	24126	4452	375	11					369056
11	78	2387	22737	100782	243012	342249	289191	145882	42372	6567	473	12				1195742
12	91	3236	36091	189706	552200	961720	1038343	700382	290356	70840	9383	586	13			3852947
13	105	4290	55289	339196	1168596	2454140	3274837	2822184	1564784	545116	113685	13052	715	14		12356003
14	120	5579	82173	580749	2330757	5786508	9304764	9908181	7019481	3271632	974688	176241	17745	861	15	39459494

Technical details

Node id: n-50e231117764dacd · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d39cf0ea889e3608

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,    2;
  2:  6,   11,    3;
  3: 10,   35,   25,     4;
  4: 15,   84,  113,    46,     5;
  5: 21,  170,  363,   276,    75,     6;
  6: 28,  307,  941,  1131,   571,   113,     7;
  7: 36,  511, 2107,  3623,  2885,  1057,   161,    8;
  8: 45,  800, 4242,  9780, 11123,  6420,  1806,  220,   9;
  9: 55, 1194, 7878, 23280, 35441, 29194, 12930, 2904, 291, 10;

%F Formula

The row polynomials satisfy 4*P(n,x) = (9+3*x-9*x^2)*P(n-3,x) + (1-3*x-3*x^2-4*x^3)*d/dx P(n-3,x) - (22+2*x+4*x^2)*P(n-2,x) - (2-6*x-8*x^2)*d/dx P(n-2,x) + (15+8*x)*P(n-1,x) + (1-4*x)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 6 + 11*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 6 + 11*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + 3*x - 9*x^2)*P[n - 3, x] + (1 - 3*x - 3*x^2 - 4*x^3)*D[P[n - 3, x], x] - (22 + 2*x + 4*x^2)*P[n - 2, x] - (2 - 6*x - 8*x^2)*D[P[n - 2, x], x] + (15 + 8*x)*P[n - 1, x] + (1 - 4*x)*D[P[n - 1, x], x])/(4)]];
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 + 2*x; v[3] = 6 + 11*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((9 + 3*x - 9*x^2)*v[i - 3] + (1 - 3*x - 3*x^2 - 4*x^3)*deriv(v[i - 3], x) - (22 + 2*x + 4*x^2)*v[i - 2] - (2 - 6*x - 8*x^2)*deriv(v[i - 2], x) + (15 + 8*x)*v[i - 1] + (1 - 4*x)*deriv(v[i - 1], x))/(4)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
  - 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 Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    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 Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    ↑ Liu–Wang theorem for three-term recurrences (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.