A166972 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-8\,n + 4\,n x - 2\,n x^{2} - 2\,n^{2} x + 3\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-13\,n x + 11\,n x^{2} - n x^{3} - 10\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,n x^{2} + 3\,n x^{3} - 3\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(-8 + 2\,x + 9\,n - 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-13\,x + x^{2} + 13\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(-3\,x^{2} + 3\,n x^{2}\right)\,P_{n-1}''(x)\)

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

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

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	10	2													12
3	83	82	3												168
4	668	2220	366	4											3258
5	5349	51964	24630	1236	5										83184
6	42798	1144734	1296984	178860	3570	6									2666952
7	342391	24552990	60464865	18543148	1013445	9366	7								104926212
8	2739136	520406264	2639571510	1608403720	190006460	4894080	23086	8							4966044264
9	21913097	10972357720	111047126172	125662813880	28293143410	1562943648	21192416	54568	9						277581544920
10	175304786	230813947866	4573553339520	9204522746472	3658612871460	377630365644	11023602240	84937512	125298	10					18056417240808
11	1402438299	4850599000906	186058121876991	647078547475560	431766409691790	76566851305164	4142997793830	69504550440	321747543	281850	11				1350534756162384
12	11219506404	101893432661604	7515083860093230	44292803048435292	47920360805167440	13810229064997848	1265135570850228	39312796956600	402859789020	1168602660	624558	12			114945233827684896
13	89756051245	2140031354047380	302284596042645666	2979233316282870380	5098648671590971275	2296795929374321208	335447397833789220	17520594658207992	333935223806475	2188981458260	4111208442	1368444	13		11032406755210746000
14	718048409974	44942992092327350	12129904406078679480	198083672967012044468	526710688926269122150	360539264259967210434	80475504026240567328	6625712980767111960	211765876608716946	2601638365080410	11308891460680	14113085556	2972242	14	1184824070114456788992

Technical details

Node id: n-3d472a94c673517d · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-b4f90a15b05d202b

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:       10,           2;
  3:       83,          82,            3;
  4:      668,        2220,          366,            4;
  5:     5349,       51964,        24630,         1236,           5;
  6:    42798,     1144734,      1296984,       178860,        3570,          6;
  7:   342391,    24552990,     60464865,     18543148,     1013445,       9366,        7;
  8:  2739136,   520406264,   2639571510,   1608403720,   190006460,    4894080,    23086,     8;
  9: 21913097, 10972357720, 111047126172, 125662813880, 28293143410, 1562943648, 21192416, 54568, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(8*n+(2*n^2-4*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - (13*n*x+(10*n^2-11*n)*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) - (3*n*x^2+(3*n^2-3*n)*x^3)*d^2/dx^2 P(n-2,x) + (9*n-8+(2*n^2-4*n+2)*x)*P(n-1,x) + ((13*n-13)*x-(n-1)*x^2)*d/dx P(n-1,x) + (3*n-3)*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[(-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*D[P[n - 2, x], {x, 2}] + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((13*n - 13)*x - (n - 1)*x^2)*D[P[n - 1, x], x] + (3*n - 3)*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] = (-(8*n + (2*n^2 - 4*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - (13*n*x + (10*n^2 - 11*n)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) - (3*n*x^2 + (3*n^2 - 3*n)*x^3)*deriv(deriv(v[i - 2], x), x) + (9*n - 8 + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((13*n - 13)*x - (n - 1)*x^2)*deriv(v[i - 1], x) + (3*n - 3)*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) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 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.