A105868 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 12\,n^{2} + 4\,n^{2} x + 13\,n^{3} - 4\,n^{3} x + 6\,n^{4} - 8\,n^{4} x + n^{5} - 2\,n^{5} x + n^{5} x^{2}\right)\,P_{n}(x) = \left(16 - 32\,x - 8\,x^{2} + 32\,n - 36\,n x + 38\,n x^{2} + 4\,n^{2} + 42\,n^{2} x + 29\,n^{2} x^{2} - 4\,n^{2} x^{3} - 28\,n^{3} + 99\,n^{3} x - 45\,n^{3} x^{2} - n^{3} x^{3} - 20\,n^{4} + 60\,n^{4} x - 44\,n^{4} x^{2} + 6\,n^{4} x^{3} - 4\,n^{5} + 11\,n^{5} x - 10\,n^{5} x^{2} + 3\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(32\,x + 8\,x^{2} - 4\,x^{3} + 96\,n x - 88\,n x^{2} + 16\,n x^{3} + 104\,n^{2} x - 170\,n^{2} x^{2} + 52\,n^{2} x^{3} - 4\,n^{2} x^{4} + 48\,n^{3} x - 100\,n^{3} x^{2} + 46\,n^{3} x^{3} - 6\,n^{3} x^{4} + 8\,n^{4} x - 18\,n^{4} x^{2} + 12\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	4														4
2	0	8	9													17
3	0	0	54	16												70
4	0	0	54	192	25											271
5	0	0	0	480	500	36										1016
6	0	0	0	320	2250	1080	49									3699
7	0	0	0	0	3500	7560	2058	64								13182
8	0	0	0	0	1750	20160	20580	3584	81							46155
9	0	0	0	0	0	22680	82320	48384	5832	100						159316
10	0	0	0	0	0	9072	154350	268800	102060	9000	121					543403
11	0	0	0	0	0	0	135828	739200	748440	198000	13310	144				1834922
12	0	0	0	0	0	0	45276	1064448	2806650	1848000	359370	19008	169			6142921
13	0	0	0	0	0	0	0	768768	5837832	9009000	4152720	617760	26364	196		20412640
14	0	0	0	0	0	0	0	219648	6810804	25225200	25435410	8648640	1015014	35672	225	67390613

Technical details

Node id: n-c72410b667de7358 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-0c57651e209b16a0

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: 0, 4;
  2: 0, 8,  9;
  3: 0, 0, 54,  16;
  4: 0, 0, 54, 192,   25;
  5: 0, 0,  0, 480,  500,    36;
  6: 0, 0,  0, 320, 2250,  1080,    49;
  7: 0, 0,  0,   0, 3500,  7560,  2058,    64;
  8: 0, 0,  0,   0, 1750, 20160, 20580,  3584,   81;
  9: 0, 0,  0,   0,    0, 22680, 82320, 48384, 5832, 100;

%F Formula

The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(2*n^5+8*n^4+4*n^3-4*n^2)*x+n^5*x^2)*P(n,x) = -(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16-(11*n^5+60*n^4+99*n^3+42*n^2-36*n-32)*x+(10*n^5+44*n^4+45*n^3-29*n^2-38*n+8)*x^2-(3*n^5+6*n^4-n^3-4*n^2)*x^3)*P(n-1,x) + ((8*n^4+48*n^3+104*n^2+96*n+32)*x-(18*n^4+100*n^3+170*n^2+88*n-8)*x^2+(12*n^4+46*n^3+52*n^2+16*n-4)*x^3-(2*n^4+6*n^3+4*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16 - (11*n^5 + 60*n^4 + 99*n^3 + 42*n^2 - 36*n - 32)*x + (10*n^5 + 44*n^4 + 45*n^3 - 29*n^2 - 38*n + 8)*x^2 - (3*n^5 + 6*n^4 - n^3 - 4*n^2)*x^3)*P[n - 1, x] + ((8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x - (18*n^4 + 100*n^3 + 170*n^2 + 88*n - 8)*x^2 + (12*n^4 + 46*n^3 + 52*n^2 + 16*n - 4)*x^3 - (2*n^4 + 6*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (2*n^5 + 8*n^4 + 4*n^3 - 4*n^2)*x + n^5*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; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16 - (11*n^5 + 60*n^4 + 99*n^3 + 42*n^2 - 36*n - 32)*x + (10*n^5 + 44*n^4 + 45*n^3 - 29*n^2 - 38*n + 8)*x^2 - (3*n^5 + 6*n^4 - n^3 - 4*n^2)*x^3)*v[i - 1] + ((8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x - (18*n^4 + 100*n^3 + 170*n^2 + 88*n - 8)*x^2 + (12*n^4 + 46*n^3 + 52*n^2 + 16*n - 4)*x^3 - (2*n^4 + 6*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (2*n^5 + 8*n^4 + 4*n^3 - 4*n^2)*x + n^5*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

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,k) = C(n,k)*C(k,n-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

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.