A209414 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,x + 12\,x^{2} + n - 3\,n x - 7\,n x^{2} + 12\,n x^{3} + n^{2} - 6\,n^{2} x + 9\,n^{2} x^{2} - 40\,n^{2} x^{3} + 3\,n^{3} x - 20\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-4\,x + 24\,x^{2} + 88\,x^{3} + 36\,x^{4} + n - n x - n x^{2} - 9\,n x^{3} - 116\,n x^{4} + n^{2} + 6\,n^{2} x - 27\,n^{2} x^{2} - 35\,n^{2} x^{3} - 8\,n^{2} x^{4} + 3\,n^{3} x - 2\,n^{3} x^{2} - 52\,n^{3} x^{3} + 96\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 11\,x^{2} - 23\,x^{3} + 11\,x^{4} + 24\,x^{5} - n x - 11\,n x^{2} - 12\,n x^{3} + 88\,n x^{4} - 64\,n x^{5} - 3\,n^{2} x^{2} - 13\,n^{2} x^{3} + 80\,n^{2} x^{4} - 64\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)

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

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	3													12
3	1	14	27	4												46
4	1	20	78	64	5											168
5	1	26	156	280	125	6										594
6	1	32	261	760	775	216	7									2052
7	1	38	393	1612	2765	1806	343	8								6966
8	1	44	552	2944	7310	8232	3724	512	9							23328
9	1	50	738	4864	16030	26676	21168	7008	729	10						77274
10	1	56	951	7480	30950	69504	82782	48672	12285	1000	11					253692
11	1	62	1191	10900	54500	155724	253638	226512	102465	20350	1331	12				826686
12	1	68	1458	15232	89515	312444	654752	808192	560439	200860	32186	1728	13			2676888
13	1	74	1752	20584	139235	576330	1489166	2387008	2307591	1276990	371228	48984	2197	14		8621154
14	1	80	2073	27064	207305	995064	3073133	6122512	7745751	6017440	2716571	653016	72163	2744	15	27634932

Technical details

Node id: n-9be05aa39608ef62 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-1a1875ec652e0428

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,   3;
  3: 1, 14,  27,    4;
  4: 1, 20,  78,   64,     5;
  5: 1, 26, 156,  280,   125,     6;
  6: 1, 32, 261,  760,   775,   216,     7;
  7: 1, 38, 393, 1612,  2765,  1806,   343,    8;
  8: 1, 44, 552, 2944,  7310,  8232,  3724,  512,   9;
  9: 1, 50, 738, 4864, 16030, 26676, 21168, 7008, 729, 10;

%F Formula

The row polynomials satisfy (n^2+n+(3*n^3-6*n^2-3*n+6)*x-(20*n^3-9*n^2+7*n-12)*x^2+(32*n^3-40*n^2+12*n)*x^3)*P(n,x) = (n^2+n+(3*n^3+6*n^2-n-4)*x-(2*n^3+27*n^2+n-24)*x^2-(52*n^3+35*n^2+9*n-88)*x^3+(96*n^3-8*n^2-116*n+36)*x^4)*P(n-1,x) - ((n+1)*x+(3*n^2+11*n+11)*x^2+(13*n^2+12*n+23)*x^3-(80*n^2+88*n+11)*x^4+(64*n^2+64*n-24)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (3*n^3 + 6*n^2 - n - 4)*x - (2*n^3 + 27*n^2 + n - 24)*x^2 - (52*n^3 + 35*n^2 + 9*n - 88)*x^3 + (96*n^3 - 8*n^2 - 116*n + 36)*x^4)*P[n - 1, x] - ((n + 1)*x + (3*n^2 + 11*n + 11)*x^2 + (13*n^2 + 12*n + 23)*x^3 - (80*n^2 + 88*n + 11)*x^4 + (64*n^2 + 64*n - 24)*x^5)*D[P[n - 1, x], x])/(n^2 + n + (3*n^3 - 6*n^2 - 3*n + 6)*x - (20*n^3 - 9*n^2 + 7*n - 12)*x^2 + (32*n^3 - 40*n^2 + 12*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + n + (3*n^3 + 6*n^2 - n - 4)*x - (2*n^3 + 27*n^2 + n - 24)*x^2 - (52*n^3 + 35*n^2 + 9*n - 88)*x^3 + (96*n^3 - 8*n^2 - 116*n + 36)*x^4)*v[i - 1] - ((n + 1)*x + (3*n^2 + 11*n + 11)*x^2 + (13*n^2 + 12*n + 23)*x^3 - (80*n^2 + 88*n + 11)*x^4 + (64*n^2 + 64*n - 24)*x^5)*deriv(v[i - 1], x))/(n^2 + n + (3*n^3 - 6*n^2 - 3*n + 6)*x - (20*n^3 - 9*n^2 + 7*n - 12)*x^2 + (32*n^3 - 40*n^2 + 12*n)*x^3)); 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 of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.?
  - 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.