A034867 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x - 2\,n^{2} x - 10\,n^{3} x + n^{3} x^{2} + 2\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(4 - 12\,x + 12\,x^{2} - 4\,x^{3} + 4\,n + 18\,n x + 26\,n x^{2} - 8\,n x^{3} - 12\,n^{2} x - 19\,n^{2} x^{2} - n^{2} x^{3} - 10\,n^{3} x - 15\,n^{3} x^{2} + 7\,n^{3} x^{3} + 3\,n^{4} x^{2} + 5\,n^{4} x^{3} + n^{5} x^{2} + n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x - 24\,x^{2} + 24\,x^{3} - 8\,x^{4} - 44\,n x^{2} + 68\,n x^{3} - 24\,n x^{4} - 20\,n^{2} x^{2} + 46\,n^{2} x^{3} - 26\,n^{2} x^{4} + 12\,n^{3} x^{3} - 12\,n^{3} x^{4} + 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	3	4			7
3	4	16			20
4	5	40	9		54
5	6	80	54		140
6	7	140	189	16	352
7	8	224	504	128	864

All 15 rows

Technical details

Node id: n-12a7b79b8c283509 · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-4b3b88e53403151e

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:  3,   4;
  3:  4,  16;
  4:  5,  40,    9;
  5:  6,  80,   54;
  6:  7, 140,  189,   16;
  7:  8, 224,  504,  128;
  8:  9, 336, 1134,  576,  25;
  9: 10, 480, 2268, 1920, 250;

%F Formula

The row polynomials satisfy (4*n-(10*n^3+2*n^2-4*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (4*n+4-(10*n^3+12*n^2-18*n+12)*x+(n^5+3*n^4-15*n^3-19*n^2+26*n+12)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) + (8*x-(20*n^2+44*n+24)*x^2+(2*n^4+12*n^3+46*n^2+68*n+24)*x^3-(2*n^4+12*n^3+26*n^2+24*n+8)*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 + 4 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] + (8*x - (20*n^2 + 44*n + 24)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n - (10*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 + n^3)*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 + 4 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] + (8*x - (20*n^2 + 44*n + 24)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n - (10*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 + n^3)*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 of odd-numbered terms in rows of Pascal's triangle.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

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.