A084534 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 6\,n + 4\,n x + 4\,n^{2} - 12\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(2 + 12\,x + 12\,x^{2} - 6\,n - 32\,n x - 32\,n x^{2} + 4\,n^{2} + 12\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + 6\,x^{2} + 8\,x^{3} - 2\,n x - 12\,n x^{2} - 16\,n x^{3} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	1	8	6						15
3	1	12	27	8					48
4	1	16	60	64	10				151
5	1	20	105	200	125	12			463
6	1	24	162	448	525	216	14		1390
7	1	28	231	840	1470	1176	343	16	4105

All 15 rows

Technical details

Node id: n-4fb262f896308a6d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b24a5168263ea116

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: 1,  4;
  2: 1,  8,   6;
  3: 1, 12,  27,    8;
  4: 1, 16,  60,   64,   10;
  5: 1, 20, 105,  200,  125,    12;
  6: 1, 24, 162,  448,  525,   216,   14;
  7: 1, 28, 231,  840, 1470,  1176,  343,   16;
  8: 1, 32, 312, 1408, 3300,  4032, 2352,  512,  18;
  9: 1, 36, 405, 2184, 6435, 10692, 9702, 4320, 729, 20;

%F Formula

The row polynomials satisfy (4*n^2-6*n+2+(8*n^3-12*n^2+4*n)*x)*P(n,x) = (4*n^2-6*n+2+(8*n^3+12*n^2-32*n+12)*x+(24*n^3-4*n^2-32*n+12)*x^2)*P(n-1,x) - ((2*n-1)*x+(4*n^2+12*n-6)*x^2+(16*n^2+16*n-8)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (8*n^3 + 12*n^2 - 32*n + 12)*x + (24*n^3 - 4*n^2 - 32*n + 12)*x^2)*P[n - 1, x] - ((2*n - 1)*x + (4*n^2 + 12*n - 6)*x^2 + (16*n^2 + 16*n - 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (8*n^3 - 12*n^2 + 4*n)*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 6*n + 2 + (8*n^3 + 12*n^2 - 32*n + 12)*x + (24*n^3 - 4*n^2 - 32*n + 12)*x^2)*v[i - 1] - ((2*n - 1)*x + (4*n^2 + 12*n - 6)*x^2 + (16*n^2 + 16*n - 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (8*n^3 - 12*n^2 + 4*n)*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 ≥ 2
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 Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-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
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A086645 → basis A013611 → θ+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023