A136394 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 6 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	1							3
3	6	5							11
4	21	26	3						50
5	85	154	35						274
6	410	999	340	15					1764
7	2366	7083	3304	315					13068
8	16065	55404	33110	4900	105				109584
9	125665	479820	347094	70532	3465				1026576
10	1112074	4588725	3848616	999530	78750	945			10628640
11	10976174	48191065	45432816	14327170	1571570	45045			120543840
12	119481285	552282422	572456577	211003716	29801695	1406790	10395		1486442880
13	1421542629	6864806870	7698323841	3223617540	556645375	37147110	675675		19802759040
14	18348340114	92034138091	110327477156	51372442121	10448696258	906670765	27747720	135135	283465647360

Technical details

Node id: n-78dcc9a0c7593b0d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e4d87171b69c1b1e

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      1;
  2:      2,      1;
  3:      6,      5;
  4:     21,     26,      3;
  5:     85,    154,     35;
  6:    410,    999,    340,    15;
  7:   2366,   7083,   3304,   315;
  8:  16065,  55404,  33110,  4900,  105;
  9: 125665, 479820, 347094, 70532, 3465;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-4*n+3-(2*n^2-8*n+6)*x+(n^2-4*n+3)*x^2)*P(n-4,x) + (2*n^2-6*n+4-(2*n^2-6*n+4)*x)*P(n-3,x) - (n^2-1-(2*n-2)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 6 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[3, x_] := 6 + 5*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*P[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] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 6 + 5*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 4] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*v[i - 3] - (n^2 - 1 - (2*n - 2)*x)*v[i - 2] + (2*n - 1)*v[i - 1]); 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 permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.