A122542 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 9\,x - x^{2} - 3\,x^{3} + 4\,x^{4} - 2\,n + 26\,n x - 20\,n x^{2} - 4\,n x^{4} + n^{2} - 22\,n^{2} x + 28\,n^{2} x^{2} - 4\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x - 16\,n^{3} x^{2} + 2\,n^{3} x^{3} + 4\,n^{4} x^{2}\right)\,P_{n}(x) = \left(x^{2} - 5\,x^{3} - 5\,x^{4} + x^{5} + 4\,n x^{3} - 2\,n x^{4} - 2\,n x^{5} + n^{2} x - 4\,n^{2} x^{2} + 4\,n^{2} x^{3} + 2\,n^{2} x^{4} + n^{2} x^{5} + 6\,n^{3} x^{2} + 2\,n^{3} x^{4} + 4\,n^{4} x^{3}\right)\,P_{n-2}(x) + \left(1 - x - 8\,x^{2} - 9\,x^{4} + x^{5} - 2\,n + 10\,n x + 28\,n x^{2} - 4\,n x^{3} + 2\,n x^{4} - 2\,n x^{5} + n^{2} - 15\,n^{2} x - 14\,n^{2} x^{2} + 18\,n^{2} x^{3} + n^{2} x^{4} + n^{2} x^{5} + 6\,n^{3} x - 2\,n^{3} x^{2} - 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1	1							1
2	1							1
3	1	8						9
4	1	16	18					35
5	1	24	72	32				129
6	1	32	162	192	50			437
7	1	40	288	608	400	72		1409
8	1	48	450	1408	1650	720	98	4375

All 15 rows

Technical details

Node id: n-abc7183f0a56451a · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-7f8284a1431ded6a

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,  8;
   4: 1, 16,  18;
   5: 1, 24,  72,   32;
   6: 1, 32, 162,  192,    50;
   7: 1, 40, 288,  608,   400,    72;
   8: 1, 48, 450, 1408,  1650,   720,   98;
   9: 1, 56, 648, 2720,  4800,  3672, 1176,  128;
  10: 1, 64, 882, 4672, 11250, 12960, 7154, 1792, 162;

%F Formula

The row polynomials satisfy (n^2-2*n+1+(6*n^3-22*n^2+26*n-9)*x+(4*n^4-16*n^3+28*n^2-20*n-1)*x^2+(2*n^3-4*n^2-3)*x^3+(n^2-4*n+4)*x^4)*P(n,x) = (n^2*x+(6*n^3-4*n^2+1)*x^2+(4*n^4+4*n^2+4*n-5)*x^3+(2*n^3+2*n^2-2*n-5)*x^4+(n^2-2*n+1)*x^5)*P(n-2,x) + (n^2-2*n+1+(6*n^3-15*n^2+10*n-1)*x+(4*n^4-2*n^3-14*n^2+28*n-8)*x^2+(4*n^4-6*n^3+18*n^2-4*n)*x^3+(2*n^3+n^2+2*n-9)*x^4+(n^2-2*n+1)*x^5)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2*x + (6*n^3 - 4*n^2 + 1)*x^2 + (4*n^4 + 4*n^2 + 4*n - 5)*x^3 + (2*n^3 + 2*n^2 - 2*n - 5)*x^4 + (n^2 - 2*n + 1)*x^5)*P[n - 2, x] + (n^2 - 2*n + 1 + (6*n^3 - 15*n^2 + 10*n - 1)*x + (4*n^4 - 2*n^3 - 14*n^2 + 28*n - 8)*x^2 + (4*n^4 - 6*n^3 + 18*n^2 - 4*n)*x^3 + (2*n^3 + n^2 + 2*n - 9)*x^4 + (n^2 - 2*n + 1)*x^5)*P[n - 1, x])/(n^2 - 2*n + 1 + (6*n^3 - 22*n^2 + 26*n - 9)*x + (4*n^4 - 16*n^3 + 28*n^2 - 20*n - 1)*x^2 + (2*n^3 - 4*n^2 - 3)*x^3 + (n^2 - 4*n + 4)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2*x + (6*n^3 - 4*n^2 + 1)*x^2 + (4*n^4 + 4*n^2 + 4*n - 5)*x^3 + (2*n^3 + 2*n^2 - 2*n - 5)*x^4 + (n^2 - 2*n + 1)*x^5)*v[i - 2] + (n^2 - 2*n + 1 + (6*n^3 - 15*n^2 + 10*n - 1)*x + (4*n^4 - 2*n^3 - 14*n^2 + 28*n - 8)*x^2 + (4*n^4 - 6*n^3 + 18*n^2 - 4*n)*x^3 + (2*n^3 + n^2 + 2*n - 9)*x^4 + (n^2 - 2*n + 1)*x^5)*v[i - 1])/(n^2 - 2*n + 1 + (6*n^3 - 22*n^2 + 26*n - 9)*x + (4*n^4 - 16*n^3 + 28*n^2 - 20*n - 1)*x^2 + (2*n^3 - 4*n^2 - 3)*x^3 + (n^2 - 4*n + 4)*x^4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 3
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 reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

Related ?

Same fingerprint ?

A266213 → (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
[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.