A122076 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20 - 2\,x + 20\,n - 18\,n x + 4\,n x^{2} + 20\,n^{2} x - 12\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-80 - 98\,x - 12\,x^{2} + 12\,x^{3} + 80\,n - 2\,n x - 92\,n x^{2} - 32\,n x^{3} + 100\,n^{2} x + 72\,n^{2} x^{2} - 4\,n^{2} x^{3} + 32\,n^{3} x^{2} + 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 50\,x - 19\,x^{2} + 14\,x^{3} + 8\,x^{4} - 50\,n x - 110\,n x^{2} - 76\,n x^{3} - 16\,n x^{4} - 20\,n^{2} x^{2} - 36\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	4														7
2	7	16	6													29
3	18	60	45	8												131
4	47	208	240	96	10											601
5	123	680	1065	680	175	12										2735
6	322	2136	4230	3728	1575	288	14									12293
7	843	6524	15624	17584	10395	3192	441	16								54619
8	2207	19520	54840	75136	57200	24864	5880	640	18							240305
9	5778	57492	185355	299352	279045	157356	53214	10080	891	20						1048583
10	15127	167240	608520	1131920	1249075	866928	382830	104640	16335	1200	22					4543837
11	39603	481624	1952115	4109864	5240675	4322472	2360281	847440	192456	25300	1573	24				19573427
12	103682	1375536	6145596	14446112	20903610	19982448	13012244	5801664	1740123	335280	37752	2016	26			83886089
13	271443	3901300	19047600	49447840	80057250	87051120	65879450	35141600	13140270	3360500	558415	54600	2535	28		357913951
14	710647	10999408	58263450	165551680	296529800	361470480	311854620	193819200	86965515	27827800	6166160	895440	76895	3136	30	1521134261

Technical details

Node id: n-d57812740ab00556 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-14b3f6a4b38fc102

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:    2;
  1:    3,     4;
  2:    7,    16,      6;
  3:   18,    60,     45,      8;
  4:   47,   208,    240,     96,     10;
  5:  123,   680,   1065,    680,    175,     12;
  6:  322,  2136,   4230,   3728,   1575,    288,    14;
  7:  843,  6524,  15624,  17584,  10395,   3192,   441,    16;
  8: 2207, 19520,  54840,  75136,  57200,  24864,  5880,   640,  18;
  9: 5778, 57492, 185355, 299352, 279045, 157356, 53214, 10080, 891, 20;

%F Formula

The row polynomials satisfy (20*n-20+(20*n^2-18*n-2)*x+(8*n^3-12*n^2+4*n)*x^2)*P(n,x) = (80*n-80+(100*n^2-2*n-98)*x+(32*n^3+72*n^2-92*n-12)*x^2+(24*n^3-4*n^2-32*n+12)*x^3)*P(n-1,x) - (25+(50*n+50)*x+(20*n^2+110*n+19)*x^2+(36*n^2+76*n-14)*x^3+(16*n^2+16*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 4*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*P[n - 1, x] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*v[i - 1] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*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 Coefficients of a generalized Jaco-Lucas polynomial (even indices) 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

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.