A274496 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 - 6\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(4 - 12\,x + 16\,x^{2} - 16\,x^{3} - 14\,n + 12\,n x - 24\,n x^{2} + 16\,n x^{3} + 6\,n^{2} - 8\,n^{2} x + 16\,n^{2} x^{2} + 4\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 24\,x^{2} - 32\,x^{3} - 32\,x^{4} - 4\,n - 12\,n x - 24\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	12	8							20
3	24	16							40
4	80	192	64						336
5	160	384	128						672
6	448	1920	2304	512					5184
7	896	3840	4608	1024					10368
8	2304	14336	30720	24576	4096				76032
9	4608	28672	61440	49152	8192				152064
10	11264	92160	286720	409600	245760	32768			1078272
11	22528	184320	573440	819200	491520	65536			2156544
12	53248	540672	2211840	4587520	4915200	2359296	262144		14929920
13	106496	1081344	4423680	9175040	9830400	4718592	524288		29859840
14	245760	2981888	15138816	41287680	64225280	55050240	22020096	2097152	203046912

Technical details

Node id: n-b6b3ea1504194976 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-4f4ca2dec4972ed6

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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:    1;
  1:    2;
  2:   12,     8;
  3:   24,    16;
  4:   80,   192,    64;
  5:  160,   384,   128;
  6:  448,  1920,  2304,   512;
  7:  896,  3840,  4608,  1024;
  8: 2304, 14336, 30720, 24576, 4096;
  9: 4608, 28672, 61440, 49152, 8192;

%F Formula

The row polynomials satisfy (n^2-n-2+(4*n-6)*x+4*x^2)*P(n,x) = (6*n^2-14*n+4+(4*n^3-8*n^2+12*n-12)*x+(16*n^2-24*n+16)*x^2+(16*n-16)*x^3)*P(n-1,x) - (4*n+(4*n^2+12*n+8)*x+(8*n^2+24*n+24)*x^2+(32*n+32)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 14*n + 4 + (4*n^3 - 8*n^2 + 12*n - 12)*x + (16*n^2 - 24*n + 16)*x^2 + (16*n - 16)*x^3)*P[n - 1, x] - (4*n + (4*n^2 + 12*n + 8)*x + (8*n^2 + 24*n + 24)*x^2 + (32*n + 32)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(n^2 - n - 2 + (4*n - 6)*x + 4*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((6*n^2 - 14*n + 4 + (4*n^3 - 8*n^2 + 12*n - 12)*x + (16*n^2 - 24*n + 16)*x^2 + (16*n - 16)*x^3)*v[i - 1] - (4*n + (4*n^2 + 12*n + 8)*x + (8*n^2 + 24*n + 24)*x^2 + (32*n + 32)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(n^2 - n - 2 + (4*n - 6)*x + 4*x^2)); 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_A013611 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; …?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

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.