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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	4862	63648	222768	319872	214200	68544	9996	576	9							904475
10	16796	251940	1046520	1860480	1627920	732564	167580	18240	810	10						5722860
11	58786	994840	4796550	10232640	11191950	6636168	2150610	369600	31185	1100	11					36463440
12	208012	3922512	21573816	53934540	71383950	53300016	22909656	5610528	751410	50600	1452	12				233646504
13	742900	15452320	95611230	274960400	429625625	390733200	212732520	69463680	13320450	1430000	78650	1872	13			1504152860
14	2674440	60843510	418746510	1364803440	2469216750	2666754090	1777836060	738840960	189729540	29279250	2576574	117936	2366	14		9721421440
15	9694845	239519700	1816357725	6628112400	13670481825	17185748580	13670481825	6986865600	2292565275	475495020	60351291	4433520	171535	2940	15	63040282096

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.