A154227 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(4\,n^{2} x^{2} + 3\,n^{3} x^{2} - 3\,n^{4} x^{2} - 3\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-16\,n x - 12\,n x^{2} - 28\,n^{2} x - 20\,n^{2} x^{2} - 16\,n^{3} x - 12\,n^{3} x^{2} - 4\,n^{4} x - 4\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-16 - 16\,x - 8\,x^{2} - 12\,n + 8\,n x - 4\,n x^{2} - 4\,n^{2} + 20\,n^{2} x - 4\,n^{2} x^{2} + 16\,n^{3} x + 4\,n^{4} x\right)\,P_{n-2}(x) + \left(24 + 16\,x + 16\,n + 8\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 1 + 16\,x + 3\,x^{2}\)

\(P_{3}(x) = 1 + 38\,x + 57\,x^{2} + 4\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	2							3
2	1	16	3						20
3	1	38	57	4					100
4	1	70	474	140	5				690
5	1	114	1776	2368	285	6			4550
6	1	172	4887	22432	8145	516	7		36160
7	1	246	11301	114196	142745	22602	861	8	291960

All 15 rows

Technical details

Node id: n-97f32c3deac53582 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1bf11212c793d813

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,   2;
  2: 1,  16,     3;
  3: 1,  38,    57,       4;
  4: 1,  70,   474,     140,       5;
  5: 1, 114,  1776,    2368,     285,        6;
  6: 1, 172,  4887,   22432,    8145,      516,       7;
  7: 1, 246, 11301,  114196,  142745,    22602,     861,      8;
  8: 1, 338, 23280,  422484, 1547290,   633726,   54320,   1352,    9;
  9: 1, 450, 44082, 1282264, 9926370, 11911644, 2243962, 117552, 2025, 10;

%F Formula

The row polynomials satisfy (4*n^2+4*n+8)*P(n,x) = -(n^6+3*n^5+3*n^4-3*n^3-4*n^2)*x^2*P(n-4,x) - ((4*n^4+16*n^3+28*n^2+16*n)*x+(4*n^4+12*n^3+20*n^2+12*n)*x^2)*P(n-3,x) - (4*n^2+12*n+16-(4*n^4+16*n^3+20*n^2+8*n-16)*x+(4*n^2+4*n+8)*x^2)*P(n-2,x) + (8*n^2+16*n+24+(8*n^2+8*n+16)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 16*x + 3*x^2, P(3,x) = 1 + 38*x + 57*x^2 + 4*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 16*x + 3*x^2;
P[3, x_] := 1 + 38*x + 57*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*P[n - 4, x] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*P[n - 3, x] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*P[n - 2, x] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*P[n - 1, x])/(4*n^2 + 4*n + 8)]];
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 + 2*x; v[3] = 1 + 16*x + 3*x^2; v[4] = 1 + 38*x + 57*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*v[i - 4] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*v[i - 3] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*v[i - 2] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*v[i - 1])/(4*n^2 + 4*n + 8)); 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 ≥ 1
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 T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.?
  - 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) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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.