A129710 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n x - 8\,n x^{2} + 2\,n^{2} - 2\,n^{2} x - 2\,n^{2} x^{2} + 2\,n^{2} x^{3} - 2\,n^{3} + 16\,n^{3} x + 11\,n^{3} x^{2} + 5\,n^{3} x^{3} - 6\,n^{4} x + 7\,n^{4} x^{2} + 4\,n^{4} x^{3} - 2\,n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n}(x) = \left(-20\,x^{2} + 8\,x^{3} - 8\,x^{4} + 4\,n x - 44\,n x^{2} - 46\,n x^{3} - 16\,n x^{4} + 2\,n^{2} + 26\,n^{2} x + 43\,n^{2} x^{2} + 11\,n^{2} x^{3} - 2\,n^{2} x^{4} - 2\,n^{3} + 48\,n^{3} x^{2} + 45\,n^{3} x^{3} + 14\,n^{3} x^{4} - 6\,n^{4} x - 13\,n^{4} x^{2} + 9\,n^{4} x^{3} + 10\,n^{4} x^{4} - 2\,n^{5} x^{2} - 3\,n^{5} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x^{2} - 8\,x^{3} - 4\,x^{4} - 16\,x^{5} - 6\,n x^{2} - 36\,n x^{3} - 60\,n x^{4} - 48\,n x^{5} + 2\,n^{2} x + 10\,n^{2} x^{2} + n^{2} x^{3} - 41\,n^{2} x^{4} - 52\,n^{2} x^{5} + 6\,n^{3} x^{2} + 27\,n^{3} x^{3} + 6\,n^{3} x^{4} - 24\,n^{3} x^{5} + 2\,n^{4} x^{3} + 7\,n^{4} x^{4} - 4\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	2	4			6
3	2	12			14
4	2	20	9		31
5	2	28	36		66
6	2	36	81	16	135
7	2	44	144	80	270

All 15 rows

Technical details

Node id: n-2b307fcb6ac49d03 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-12a125bb843728b5

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: 2;
  2: 2,  4;
  3: 2, 12;
  4: 2, 20,   9;
  5: 2, 28,  36;
  6: 2, 36,  81,  16;
  7: 2, 44, 144,  80;
  8: 2, 52, 225, 224,  25;
  9: 2, 60, 324, 480, 150;

%F Formula

The row polynomials satisfy -(2*n^3-2*n^2+(6*n^4-16*n^3+2*n^2-4*n)*x+(2*n^5-7*n^4-11*n^3+2*n^2+8*n)*x^2-(n^5+4*n^4+5*n^3+2*n^2)*x^3)*P(n,x) = -(2*n^3-2*n^2+(6*n^4-26*n^2-4*n)*x+(2*n^5+13*n^4-48*n^3-43*n^2+44*n+20)*x^2+(3*n^5-9*n^4-45*n^3-11*n^2+46*n-8)*x^3-(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^4)*P(n-1,x) + (2*n^2*x+(6*n^3+10*n^2-6*n-2)*x^2+(2*n^4+27*n^3+n^2-36*n-8)*x^3+(7*n^4+6*n^3-41*n^2-60*n-4)*x^4-(4*n^4+24*n^3+52*n^2+48*n+16)*x^5)*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[(-(2*n^3 - 2*n^2 + (6*n^4 - 26*n^2 - 4*n)*x + (2*n^5 + 13*n^4 - 48*n^3 - 43*n^2 + 44*n + 20)*x^2 + (3*n^5 - 9*n^4 - 45*n^3 - 11*n^2 + 46*n - 8)*x^3 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^4)*P[n - 1, x] + (2*n^2*x + (6*n^3 + 10*n^2 - 6*n - 2)*x^2 + (2*n^4 + 27*n^3 + n^2 - 36*n - 8)*x^3 + (7*n^4 + 6*n^3 - 41*n^2 - 60*n - 4)*x^4 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 2*n^2 - (6*n^4 - 16*n^3 + 2*n^2 - 4*n)*x - (2*n^5 - 7*n^4 - 11*n^3 + 2*n^2 + 8*n)*x^2 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^3)]];
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] = (-(2*n^3 - 2*n^2 + (6*n^4 - 26*n^2 - 4*n)*x + (2*n^5 + 13*n^4 - 48*n^3 - 43*n^2 + 44*n + 20)*x^2 + (3*n^5 - 9*n^4 - 45*n^3 - 11*n^2 + 46*n - 8)*x^3 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^4)*v[i - 1] + (2*n^2*x + (6*n^3 + 10*n^2 - 6*n - 2)*x^2 + (2*n^4 + 27*n^3 + n^2 - 36*n - 8)*x^3 + (7*n^4 + 6*n^3 - 41*n^2 - 60*n - 4)*x^4 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^5)*deriv(v[i - 1], x))/(-2*n^3 + 2*n^2 - (6*n^4 - 16*n^3 + 2*n^2 - 4*n)*x - (2*n^5 - 7*n^4 - 11*n^3 + 2*n^2 + 8*n)*x^2 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^3)); 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) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= f…?
  - 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.