A358735 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-6\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 12\,n - 8\,n x + 2\,n x^{2} - 6\,n^{2} + 12\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 10\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 1 + 8\,x + 6\,x^{2}\)

\(P_{3}(x) = 1 + 20\,x + 48\,x^{2} + 24\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	1	8	6													15
3	1	20	48	24												93
4	1	40	210	304	120											675
5	1	70	672	1984	2140	720										5587
6	1	112	1764	9040	19040	16848	5040									51845
7	1	168	4032	32560	115040	192912	147672	40320								532705
8	1	240	8316	99088	536640	1473360	2090816	1430784	362880							6002125
9	1	330	15840	265408	2072980	8549520	19504240	24326784	15212448	3628800						73576351
10	1	440	28314	642928	6926920	40501440	136662064	270128384	303738336	176284800	39916800					974830427
11	1	572	48048	1435720	20621120	163888704	773588368	2233967360	3935102976	4061298240	2212879680	479001600				13881832389
12	1	728	78078	2997280	55851120	584220096	3704956528	14807018240	37725026976	60414351360	57994979328	29926609920	6227020800			211443110455
13	1	910	122304	5912192	139786920	1875828960	15496996144	82425137536	288067592736	661684460800	977702303424	881848470528	433920464640	87178291200		3430345368295
14	1	1120	185640	11110112	327186080	5515858080	57927575440	397995195136	1836606074688	5744006012800	12087350323840	16665950965248	14236315711488	6716542049280	1307674368000	59056222616953

Technical details

Node id: n-67fe0ee394103768 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f387468580877db5

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,   8,     6;
  3: 1,  20,    48,     24;
  4: 1,  40,   210,    304,     120;
  5: 1,  70,   672,   1984,    2140,     720;
  6: 1, 112,  1764,   9040,   19040,   16848,     5040;
  7: 1, 168,  4032,  32560,  115040,  192912,   147672,    40320;
  8: 1, 240,  8316,  99088,  536640, 1473360,  2090816,  1430784,   362880;
  9: 1, 330, 15840, 265408, 2072980, 8549520, 19504240, 24326784, 15212448, 3628800;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 8*x + 6*x^2, P(3,x) = 1 + 20*x + 48*x^2 + 24*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 8*x + 6*x^2;
P[3, x_] := 1 + 20*x + 48*x^2 + 24*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
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 + 8*x + 6*x^2; v[4] = 1 + 20*x + 48*x^2 + 24*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 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
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 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] 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
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023