A280580 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 16\,x - 16\,n - 20\,n x + 8\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 5\,x^{2} + 16\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	6	2						8
4	30	30	3					63
5	140	280	84	4				508
6	630	2100	1260	180	5			4175
7	2772	13860	13860	3960	330	6		34788
8	12012	84084	126126	60060	10010	546	7	292845

All 15 rows

Technical details

Node id: n-89b33e7a8df4304e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c613f7c62d85a96e

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 -2)
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:      0;
   2:      1;
   3:      6,       2;
   4:     30,      30,       3;
   5:    140,     280,      84,       4;
   6:    630,    2100,    1260,     180,       5;
   7:   2772,   13860,   13860,    3960,     330,      6;
   8:  12012,   84084,  126126,   60060,   10010,    546,     7;
   9:  51480,  480480, 1009008,  720720,  200200,  21840,   840,    8;
  10: 218790, 2625480, 7351344, 7351344, 3063060, 556920, 42840, 1224, 9;

%F Formula

The row polynomials satisfy (2*n^2-5*n+2)*P(n,x) = (8*n^2-16*n+6+(6*n^2-20*n+16)*x)*P(n-1,x) + ((16*n-20)*x-(4*n-5)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 16*n + 6 + (6*n^2 - 20*n + 16)*x)*P[n - 1, x] + ((16*n - 20)*x - (4*n - 5)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 16*n + 6 + (6*n^2 - 20*n + 16)*x)*v[i - 1] + ((16*n - 20)*x - (4*n - 5)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 2)); 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 Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

References

[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.