A084534 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2 + 4\,x + 10\,n - 36\,n x - 16\,n^{2} + 104\,n^{2} x - 8\,n^{2} x^{2} + 8\,n^{3} - 120\,n^{3} x + 40\,n^{3} x^{2} + 48\,n^{4} x - 64\,n^{4} x^{2} + 32\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-2 - 20\,x - 60\,x^{2} - 24\,x^{3} + 10\,n + 84\,n x + 192\,n x^{2} + 88\,n x^{3} - 16\,n^{2} - 88\,n^{2} x - 76\,n^{2} x^{2} - 24\,n^{2} x^{3} + 8\,n^{3} - 24\,n^{3} x - 264\,n^{3} x^{2} - 200\,n^{3} x^{3} + 48\,n^{4} x + 176\,n^{4} x^{2} + 64\,n^{4} x^{3} + 32\,n^{5} x^{2} + 96\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-x - 10\,x^{2} - 28\,x^{3} - 16\,x^{4} + 4\,n x + 32\,n x^{2} + 76\,n x^{3} + 48\,n x^{4} - 4\,n^{2} x - 16\,n^{2} x^{2} + 4\,n^{2} x^{3} + 16\,n^{2} x^{4} - 24\,n^{3} x^{2} - 128\,n^{3} x^{3} - 128\,n^{3} x^{4} - 16\,n^{4} x^{3} - 64\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 1 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	8							9
2	1	16	18						35
3	1	24	81	32					138
4	1	32	180	256	50				519
5	1	40	315	800	625	72			1853
6	1	48	486	1792	2625	1296	98		6346
7	1	56	693	3360	7350	7056	2401	128	21045

All 15 rows

Technical details

Node id: n-08fd024edc5659fd · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-31dd68f9934466a8

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,  8;
  2: 1, 16,   18;
  3: 1, 24,   81,   32;
  4: 1, 32,  180,  256,    50;
  5: 1, 40,  315,  800,   625,    72;
  6: 1, 48,  486, 1792,  2625,  1296,    98;
  7: 1, 56,  693, 3360,  7350,  7056,  2401,   128;
  8: 1, 64,  936, 5632, 16500, 24192, 16464,  4096,  162;
  9: 1, 72, 1215, 8736, 32175, 64152, 67914, 34560, 6561, 200;

%F Formula

The row polynomials satisfy (8*n^3-16*n^2+10*n-2+(48*n^4-120*n^3+104*n^2-36*n+4)*x+(32*n^5-64*n^4+40*n^3-8*n^2)*x^2)*P(n,x) = (8*n^3-16*n^2+10*n-2+(48*n^4-24*n^3-88*n^2+84*n-20)*x+(32*n^5+176*n^4-264*n^3-76*n^2+192*n-60)*x^2+(96*n^5+64*n^4-200*n^3-24*n^2+88*n-24)*x^3)*P(n-1,x) - ((4*n^2-4*n+1)*x+(24*n^3+16*n^2-32*n+10)*x^2+(16*n^4+128*n^3-4*n^2-76*n+28)*x^3+(64*n^4+128*n^3-16*n^2-48*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 24*n^3 - 88*n^2 + 84*n - 20)*x + (32*n^5 + 176*n^4 - 264*n^3 - 76*n^2 + 192*n - 60)*x^2 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^3)*P[n - 1, x] - ((4*n^2 - 4*n + 1)*x + (24*n^3 + 16*n^2 - 32*n + 10)*x^2 + (16*n^4 + 128*n^3 - 4*n^2 - 76*n + 28)*x^3 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 120*n^3 + 104*n^2 - 36*n + 4)*x + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^2)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 24*n^3 - 88*n^2 + 84*n - 20)*x + (32*n^5 + 176*n^4 - 264*n^3 - 76*n^2 + 192*n - 60)*x^2 + (96*n^5 + 64*n^4 - 200*n^3 - 24*n^2 + 88*n - 24)*x^3)*v[i - 1] - ((4*n^2 - 4*n + 1)*x + (24*n^3 + 16*n^2 - 32*n + 10)*x^2 + (16*n^4 + 128*n^3 - 4*n^2 - 76*n + 28)*x^3 + (64*n^4 + 128*n^3 - 16*n^2 - 48*n + 16)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 16*n^2 + 10*n - 2 + (48*n^4 - 120*n^3 + 104*n^2 - 36*n + 4)*x + (32*n^5 - 64*n^4 + 40*n^3 - 8*n^2)*x^2)); 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)²

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