A228955 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	6							6
3	36	12						48
4	240	240						480
5	1800	3600	360					5760
6	15120	50400	15120					80640
7	141120	705600	423360	20160				1290240
8	1451520	10160640	10160640	1451520				23224320
9	16329600	152409600	228614400	65318400	1814400			464486400
10	199584000	2395008000	5029516800	2395008000	199584000			10218700800
11	2634508800	39517632000	110649369600	79035264000	13172544000	239500800		245248819200
12	37362124800	684972288000	2465900236800	2465900236800	684972288000	37362124800		6376469299200
13	566658892800	12466495641600	56099230387200	74798973849600	31166239104000	3399953356800	43589145600	178541140377600
14	9153720576000	237996734976000	1308982042368000	2243969215488000	1308982042368000	237996734976000	9153720576000	5356234211328000

Technical details

Node id: n-131f59264e77673e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-937db0732045a112

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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;
  3:       36,        12;
  4:      240,       240;
  5:     1800,      3600,       360;
  6:    15120,     50400,     15120;
  7:   141120,    705600,    423360,    20160;
  8:  1451520,  10160640,  10160640,  1451520;
  9: 16329600, 152409600, 228614400, 65318400, 1814400;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n^2+n+(n^2-n-2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx 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[((n^2 + n + (n^2 - n - 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(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] = ((n^2 + n + (n^2 - n - 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], 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 Table: T(n,k) = n!*binomial(n+1,2*k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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.