A120986 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 2\,n^{2} - 2\,n^{2} x + 4\,n^{3} - 2\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(-8\,x - 4\,x^{2} + 8\,n + 38\,n x + 14\,n x^{2} - 22\,n^{2} - 43\,n^{2} x - 10\,n^{2} x^{2} + 14\,n^{3} - 2\,n^{3} x - 12\,n^{3} x^{2} + 15\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3} - n - 8\,n x + 3\,n x^{2} + 6\,n x^{3} + n^{2} + 5\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + n^{3} x + 7\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	2							4
3	5	12	3						20
4	14	56	36	4					110
5	42	240	270	80	5				637
6	132	990	1650	880	150	6			3808
7	429	4004	9009	7280	2275	252	7		23256
8	1430	16016	45864	50960	24500	5040	392	8	144210

All 15 rows

Technical details

Node id: n-a16f92cc711fac02 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-dfc7b142e94d834e

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     2,      2;
   3:     5,     12,       3;
   4:    14,     56,      36,       4;
   5:    42,    240,     270,      80,       5;
   6:   132,    990,    1650,     880,     150,      6;
   7:   429,   4004,    9009,    7280,    2275,    252,      7;
   8:  1430,  16016,   45864,   50960,   24500,   5040,    392,     8;
   9:  4862,  63648,  222768,  319872,  214200,  68544,   9996,   576,   9;
  10: 16796, 251940, 1046520, 1860480, 1627920, 732564, 167580, 18240, 810, 10;

%F Formula

The row polynomials satisfy (4*n^3-2*n^2-2*n+(4*n^4-2*n^3-2*n^2)*x)*P(n,x) = (14*n^3-22*n^2+8*n+(15*n^4-2*n^3-43*n^2+38*n-8)*x+(12*n^4-12*n^3-10*n^2+14*n-4)*x^2)*P(n-1,x) + (n^2-n+(n^3+5*n^2-8*n+2)*x+(7*n^3-4*n^2+3*n)*x^2-(8*n^3+2*n^2-6*n+2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^3 - 22*n^2 + 8*n + (15*n^4 - 2*n^3 - 43*n^2 + 38*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*P[n - 1, x] + (n^2 - n + (n^3 + 5*n^2 - 8*n + 2)*x + (7*n^3 - 4*n^2 + 3*n)*x^2 - (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((14*n^3 - 22*n^2 + 8*n + (15*n^4 - 2*n^3 - 43*n^2 + 38*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*v[i - 1] + (n^2 - n + (n^3 + 5*n^2 - 8*n + 2)*x + (7*n^3 - 4*n^2 + 3*n)*x^2 - (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.