A088617 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + n x - n x^{2} - 3\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(-2\,x + 2\,x^{3} + 2\,n + 5\,n x - n x^{2} - 2\,n x^{3} - n^{2} - 4\,n^{2} x + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(4\,x + 2\,x^{2} - 4\,x^{3} - 2\,x^{4} - 4\,n x - 4\,n x^{2} + 2\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	4	7	2													13
4	7	26	25	5												63
5	11	70	135	91	14											321
6	16	155	490	644	336	42										1683
7	22	301	1400	2940	2940	1254	132									8989
8	29	532	3402	10290	16170	13068	4719	429								48639
9	37	876	7350	30030	66990	84084	57057	17875	1430							265729
10	46	1365	14520	76692	228228	402402	420420	245960	68068	4862						1462563
11	56	2035	26730	176748	672672	1567566	2282280	2042040	1050192	260338	16796					8097453
12	67	2926	46475	375375	1771770	5237232	10005996	12398100	9699690	4450940	999362	58786				45046719
13	79	4082	77077	745745	4264260	15519504	37369332	60415212	65126490	45265220	18752734	3848222	208012			251595969
14	92	5551	122850	1401400	9529520	41745132	123047496	249420600	349188840	333022690	208220012	78628536	14858000	742900		1409933619
15	106	7385	189280	2512328	20011992	103648116	365816880	903456840	1579109532	1948344398	1665760096	946454600	328361800	57500460	2674440	7923848253

Technical details

Node id: n-cc0179e01d57c43d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-695598672c6e8bd2

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,    1;
   3:  4,    7,     2;
   4:  7,   26,    25,     5;
   5: 11,   70,   135,    91,     14;
   6: 16,  155,   490,   644,    336,     42;
   7: 22,  301,  1400,  2940,   2940,   1254,    132;
   8: 29,  532,  3402, 10290,  16170,  13068,   4719,    429;
   9: 37,  876,  7350, 30030,  66990,  84084,  57057,  17875,  1430;
  10: 46, 1365, 14520, 76692, 228228, 402402, 420420, 245960, 68068, 4862;

%F Formula

The row polynomials satisfy (n^3-3*n^2+4*n+n*x-n*x^2)*P(n,x) = (n^3-n^2+2*n+(2*n^3-4*n^2+5*n-2)*x-n*x^2-(2*n-2)*x^3)*P(n-1,x) + ((2*n^2-4*n+4)*x+(2*n^2-4*n+2)*x^2-4*x^3-2*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[((n^3 - n^2 + 2*n + (2*n^3 - 4*n^2 + 5*n - 2)*x - n*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + ((2*n^2 - 4*n + 4)*x + (2*n^2 - 4*n + 2)*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 4*n + n*x - 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] = ((n^3 - n^2 + 2*n + (2*n^3 - 4*n^2 + 5*n - 2)*x - n*x^2 - (2*n - 2)*x^3)*v[i - 1] + ((2*n^2 - 4*n + 4)*x + (2*n^2 - 4*n + 2)*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 4*n + n*x - 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) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - 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

Related ?

Same fingerprint ?

A253284 → D+1

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.