A133080 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 - 6\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(16 + 8\,x - 44\,n - 44\,n x + 30\,n^{2} + 36\,n^{2} x - 6\,n^{3} - 8\,n^{3} x\right)\,P_{n-1}(x) + \left(3 + 8\,x + 4\,x^{2} - 7\,n - 22\,n x - 16\,n x^{2} + 3\,n^{2} + 10\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	4	8													12
3	10	32	24												66
4	8	48	96	64											216
5	18	128	336	384	160										1026
6	12	120	480	960	960	384									2916
7	26	288	1320	3200	4320	3072	896								13122
8	16	224	1344	4480	8960	10752	7168	2048							34992
9	34	512	3360	12544	29120	43008	39424	20480	4608						153090
10	20	360	2880	13440	40320	80640	107520	92160	46080	10240					393660
11	42	800	6840	34560	114240	258048	403200	430080	299520	122880	22528				1692738
12	24	528	5280	31680	126720	354816	709632	1013760	1013760	675840	270336	49152			4251528
13	50	1152	12144	77440	332640	1013760	2247168	3649536	4308480	3604480	2027520	688128	106496		18068994
14	28	728	8736	64064	320320	1153152	3075072	6150144	9225216	10250240	8200192	4472832	1490944	229376	44641044

Technical details

Node id: n-cab7edd027ca4c46 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-afd7cfa5f84abe43

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  2;
  2:  4,   8;
  3: 10,  32,   24;
  4:  8,  48,   96,    64;
  5: 18, 128,  336,   384,   160;
  6: 12, 120,  480,   960,   960,   384;
  7: 26, 288, 1320,  3200,  4320,  3072,   896;
  8: 16, 224, 1344,  4480,  8960, 10752,  7168,  2048;
  9: 34, 512, 3360, 12544, 29120, 43008, 39424, 20480, 4608;

%F Formula

The row polynomials satisfy (2*n^2-6*n+4)*P(n,x) = -(6*n^3-30*n^2+44*n-16+(8*n^3-36*n^2+44*n-8)*x)*P(n-1,x) + (3*n^2-7*n+3+(10*n^2-22*n+8)*x+(8*n^2-16*n+4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 30*n^2 + 44*n - 16 + (8*n^3 - 36*n^2 + 44*n - 8)*x)*P[n - 1, x] + (3*n^2 - 7*n + 3 + (10*n^2 - 22*n + 8)*x + (8*n^2 - 16*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4)]];
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] = 2; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(6*n^3 - 30*n^2 + 44*n - 16 + (8*n^3 - 36*n^2 + 44*n - 8)*x)*v[i - 1] + (3*n^2 - 7*n + 3 + (10*n^2 - 22*n + 8)*x + (8*n^2 - 16*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4)); 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 basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    Parent-row provers of polytool polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.