A120986 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n - 10\,n x + 2\,n x^{2} + 8\,n^{2} - 16\,n^{2} x + 4\,n^{2} x^{2} - 6\,n^{3} + 8\,n^{3} x + 4\,n^{4}\right)\,P_{n}(x) = \left(8 - 10\,x - 14\,x^{2} + 4\,x^{3} - 32\,n + 47\,n x + 61\,n x^{2} - 16\,n x^{3} + 43\,n^{2} - 65\,n^{2} x - 65\,n^{2} x^{2} + 12\,n^{2} x^{3} - 22\,n^{3} - 2\,n^{3} x + 24\,n^{3} x^{2} + 15\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x) + \left(-3 + 7\,x + x^{2} - 7\,x^{3} + 2\,x^{4} + 5\,n - 7\,n x - 25\,n x^{2} + 35\,n x^{3} - 8\,n x^{4} - 3\,n^{2} - n^{2} x + 20\,n^{2} x^{2} - 16\,n^{2} x^{3} + n^{3} + 7\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	1														4
3	11	8	1													20
4	42	52	15	1												110
5	162	300	150	24	1											637
6	627	1595	1210	340	35	1										3808
7	2431	8008	8463	3640	665	48	1									23256
8	9438	38584	53508	32340	9100	1176	63	1								144210
9	36686	180336	314160	251328	99960	19992	1932	80	1							904475
10	142766	823650	1744200	1767456	936054	265734	39900	3000	99	1						5722860
11	556206	3695120	9273330	11511720	7768530	2949408	630630	73920	4455	120	1					36463440
12	2169268	16343800	47642177	70590795	58693470	28520184	8182020	1369236	129030	6380	143	1				233646504
13	8469060	71466980	238090710	412440600	411536125	247464360	91171080	20523360	2767050	214500	8866	168	1			1504152860
14	33096195	309586095	1163184750	2316574260	2716138425	1968318495	900462420	261003600	47432385	5270265	342342	12012	195	1		9721421440
15	129454695	1330665000	5576536875	12593413560	17055553515	14581847280	8066433375	2911194000	682674993	102414312	9550541	527800	15925	224	1	63040282096

Technical details

Node id: n-566ab203e6213215 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c83ca58a92c3dde7

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:      3,      1;
   3:     11,      8,       1;
   4:     42,     52,      15,       1;
   5:    162,    300,     150,      24,      1;
   6:    627,   1595,    1210,     340,     35,      1;
   7:   2431,   8008,    8463,    3640,    665,     48,     1;
   8:   9438,  38584,   53508,   32340,   9100,   1176,    63,    1;
   9:  36686, 180336,  314160,  251328,  99960,  19992,  1932,   80,  1;
  10: 142766, 823650, 1744200, 1767456, 936054, 265734, 39900, 3000, 99, 1;

%F Formula

The row polynomials satisfy (4*n^4-6*n^3+8*n^2+6*n+(8*n^3-16*n^2-10*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (15*n^4-22*n^3+43*n^2-32*n+8+(12*n^4-2*n^3-65*n^2+47*n-10)*x+(24*n^3-65*n^2+61*n-14)*x^2+(12*n^2-16*n+4)*x^3)*P(n-1,x) + (n^3-3*n^2+5*n-3+(7*n^3-n^2-7*n+7)*x-(8*n^3-20*n^2+25*n-1)*x^2-(16*n^2-35*n+7)*x^3-(8*n-2)*x^4)*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[((15*n^4 - 22*n^3 + 43*n^2 - 32*n + 8 + (12*n^4 - 2*n^3 - 65*n^2 + 47*n - 10)*x + (24*n^3 - 65*n^2 + 61*n - 14)*x^2 + (12*n^2 - 16*n + 4)*x^3)*P[n - 1, x] + (n^3 - 3*n^2 + 5*n - 3 + (7*n^3 - n^2 - 7*n + 7)*x - (8*n^3 - 20*n^2 + 25*n - 1)*x^2 - (16*n^2 - 35*n + 7)*x^3 - (8*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 6*n^3 + 8*n^2 + 6*n + (8*n^3 - 16*n^2 - 10*n)*x + (4*n^2 + 2*n)*x^2)]];
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] = ((15*n^4 - 22*n^3 + 43*n^2 - 32*n + 8 + (12*n^4 - 2*n^3 - 65*n^2 + 47*n - 10)*x + (24*n^3 - 65*n^2 + 61*n - 14)*x^2 + (12*n^2 - 16*n + 4)*x^3)*v[i - 1] + (n^3 - 3*n^2 + 5*n - 3 + (7*n^3 - n^2 - 7*n + 7)*x - (8*n^3 - 20*n^2 + 25*n - 1)*x^2 - (16*n^2 - 35*n + 7)*x^3 - (8*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 6*n^3 + 8*n^2 + 6*n + (8*n^3 - 16*n^2 - 10*n)*x + (4*n^2 + 2*n)*x^2)); 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.