A166960 → D

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 0\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	6	2													8
3	27	42	3												72
4	112	540	174	4											830
5	453	5756	5214	564	5										11992
6	1818	56334	117960	34956	1590	6									212664
7	7279	528822	2309865	1503020	188325	4110	7								4541428
8	29124	4861304	41717238	52974600	14294800	879360	10038	8							114766472
9	116505	44217720	718519500	1658102904	845704050	111703680	3720192	23592	9						3382108152
10	466030	400056570	12026924640	48158754600	42881117580	10547704620	759376128	14665896	53982	10					114789120056
11	1864131	3609829730	197831557935	1332256061160	1965109335630	825581572956	110128349142	4659371304	54895599	121170	11				4439232958768
12	7456536	32529478452	3219452372910	35680380224220	84066496694280	56960415494136	12827285355780	1006713927864	26456730768	197646900	268158	12			193819935650016
13	29826157	292944262932	52047974361018	934935537244300	3427798158516555	3598098727477752	1286122989305364	169502356310904	8308438921035	141500744980	690849522	587100	13		9477249348407632
14	119304642	2637273846470	838040586509064	24136758721735764	135087427922149730	213327057758388642	115894126105335648	24018674775626136	1974984797202102	63279584148330	722085609080	2360181972	1274598	14	515343712091312192

Technical details

Node id: n-c1338451c8a75e40 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-15d50c9a2b2cc2b4

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      0;
  1:      1;
  2:      6,        2;
  3:     27,       42,         3;
  4:    112,      540,       174,          4;
  5:    453,     5756,      5214,        564,         5;
  6:   1818,    56334,    117960,      34956,      1590,         6;
  7:   7279,   528822,   2309865,    1503020,    188325,      4110,       7;
  8:  29124,  4861304,  41717238,   52974600,  14294800,    879360,   10038,     8;
  9: 116505, 44217720, 718519500, 1658102904, 845704050, 111703680, 3720192, 23592, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(4*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (5*n*x+(4*n^2-5*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (n*x^2+(n^2-n)*x^3)*d^2/dx^2 P(n-2,x) + (5*n-4+(2*n^2-4*n+2)*x)*P(n-1,x) + ((5*n-5)*x-(n-1)*x^2)*d/dx P(n-1,x) + (n-1)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (n*x^2 + (n^2 - n)*x^3)*D[P[n - 2, x], {x, 2}] + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((5*n - 5)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (n - 1)*x^2*D[P[n - 1, x], {x, 2}])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (5*n*x + (4*n^2 - 5*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (n*x^2 + (n^2 - n)*x^3)*deriv(deriv(v[i - 2], x), x) + (5*n - 4 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((5*n - 5)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.?
  - 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.