A071951 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 8\,x + 2\,x^{2} - x^{3}\right)\,P_{n-2}(x) + \left(-12\,x + 15\,x^{2} - 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 3\,x^{3} - x^{4}\right)\,P_{n-2}''(x) + \left(8 - 4\,x + x^{2}\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)

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

\(P_{2}(x) = 3 + x\)

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	12	10	1													23
4	60	92	23	1												176
5	336	904	412	44	1											1697
6	1968	9584	7100	1372	75	1										20100
7	11712	107296	124816	38432	3752	118	1									286127
8	70080	1241024	2268848	1046800	160904	8904	175	1								4796736
9	420096	14612608	42519232	28622912	6469104	556880	18984	248	1							93220065
10	2519808	173672192	815322560	794938048	254083216	31764656	1664168	37224	339	1						2074002212
11	15117312	2073989632	15881088256	22474225664	9949287424	1730885664	130253552	4426400	68244	450	1					52259342599
12	90700800	24827411456	312489698048	645552903424	392154190848	92401126144	9566111568	462123376	10711844	118404	583	1				1477555096496
13	544198656	297566144512	6188042877952	18778712775680	15615869664512	4901599455232	678046382080	44620003584	1456170144	23974808	196196	740	1			46506481844097
14	3265179648	3568616960000	123018846878720	551430830439424	628540016129792	260401776088320	47179183224832	4106304039424	180909377376	4156893312	50253060	312676	923	1		1618433955777508
15	19591053312	42810342842368	2451466823274496	16302235247108096	25544809880918016	13908997637877248	3256691189439744	367029485038592	21199629434752	652180621248	10916894560	99616192	481936	1134	1	61895923024601695

Technical details

Node id: n-917f61428ed30092 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6fd4666ecad1df2d

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:      12,        10,         1;
   4:      60,        92,        23,         1;
   5:     336,       904,       412,        44,         1;
   6:    1968,      9584,      7100,      1372,        75,        1;
   7:   11712,    107296,    124816,     38432,      3752,      118,       1;
   8:   70080,   1241024,   2268848,   1046800,    160904,     8904,     175,     1;
   9:  420096,  14612608,  42519232,  28622912,   6469104,   556880,   18984,   248,   1;
  10: 2519808, 173672192, 815322560, 794938048, 254083216, 31764656, 1664168, 37224, 339, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(12 - 8*x - 2*x^2 + x^3)*v[i - 2] - (12*x - 15*x^2 + 4*x^3)*deriv(v[i - 2], x) - (2*x^2 - 3*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (8 - 4*x + x^2)*v[i - 1] + 6*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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 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 of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.