A322944 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	1							3
3	8	8	1						17
4	44	68	18	1					131
5	296	648	272	32	1				1249
6	2320	6880	4080	760	50	1			14091
7	20640	80480	63400	16600	1720	72	1		182913
8	204960	1027040	1034320	356720	51660	3388	98	1	2678187

All 15 rows

Technical details

Node id: n-8b6beceada871ce2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f93545dd51c2bf28

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:        8,         8,         1;
   4:       44,        68,        18,         1;
   5:      296,       648,       272,        32,        1;
   6:     2320,      6880,      4080,       760,       50,       1;
   7:    20640,     80480,     63400,     16600,     1720,      72,      1;
   8:   204960,   1027040,   1034320,    356720,    51660,    3388,     98,     1;
   9:  2244480,  14183680,  17783360,   7754880,  1470000,  134064,   6048,   128,   1;
  10: 26853120, 210591360, 322304640, 172773440, 41123040, 4884096, 305088, 10032, 162, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(n^2 - n - 2 - (n - 2)*x - 2*x^2)*v[i - 2] - ((3*n - 6)*x - 6*x^2)*deriv(v[i - 2], x) + (2*n - x)*v[i - 1] + 3*x*deriv(v[i - 1], 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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.