A079510 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(x^{2} - n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(2\,x + 2\,n x - n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 2\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4\,x\)

\(P_{2}(x) = 6\,x + 18\,x^{2}\)

\(P_{3}(x) = 60\,x^{2} + 96\,x^{3}\)

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	6	18													24
3	0	0	60	96												156
4	0	0	45	520	600											1165
5	0	0	0	840	4620	4320										9780
6	0	0	0	420	11900	43848	35280									91448
7	0	0	0	0	12600	158592	449568	322560								943320
8	0	0	0	0	4725	264600	2125620	4987008	3265920							10647873
9	0	0	0	0	0	207900	4937240	29430720	59732640	36288000						130596500
10	0	0	0	0	0	62370	6063750	88790240	425919384	769982400	439084800					1729902944
11	0	0	0	0	0	0	3783780	150870720	1595982960	6475360320	10642769280	5748019200				24616786260
12	0	0	0	0	0	0	945945	146306160	3505041540	29205256080	103608400896	157160839680	80951270400			374578060701
13	0	0	0	0	0	0	0	75675600	4680535860	79349270000	549481492560	1744926962688	2470784359680	1220496076800		6069794373188
14	0	0	0	0	0	0	0	16216200	3745942200	136426290000	1791149680320	10686925529280	30908926788480	41220344793600	19615115520000	104362650760080

Technical details

Node id: n-5bec3bef59f9aa65 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-bf7964c6060fd8e6

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, 6, 18;
  3: 0, 0, 60,  96;
  4: 0, 0, 45, 520,   600;
  5: 0, 0,  0, 840,  4620,   4320;
  6: 0, 0,  0, 420, 11900,  43848,   35280;
  7: 0, 0,  0,   0, 12600, 158592,  449568,   322560;
  8: 0, 0,  0,   0,  4725, 264600, 2125620,  4987008,  3265920;
  9: 0, 0,  0,   0,     0, 207900, 4937240, 29430720, 59732640, 36288000;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-4,x) - (2*n^2+n-1)*x^2*P(n-3,x) + ((2*n+2)*x-(n^2+n)*x^2)*P(n-2,x) + (2*n+2)*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4*x, P(2,x) = 6*x + 18*x^2, P(3,x) = 60*x^2 + 96*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4*x;
P[2, x_] := 6*x + 18*x^2;
P[3, x_] := 60*x^2 + 96*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 4, x] - (2*n^2 + n - 1)*x^2*P[n - 3, x] + ((2*n + 2)*x - (n^2 + n)*x^2)*P[n - 2, x] + (2*n + 2)*x*P[n - 1, x]];
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] = 4*x; v[3] = 6*x + 18*x^2; v[4] = 60*x^2 + 96*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 4] - (2*n^2 + n - 1)*x^2*v[i - 3] + ((2*n + 2)*x - (n^2 + n)*x^2)*v[i - 2] + (2*n + 2)*x*v[i - 1]); 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 T(n,k) read by rows; related to number of preorders.?
  - 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.