A274496 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,x + 2\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(4 - 10\,x - 24\,x^{2} - 8\,x^{3} + 20\,n x + 10\,n x^{2} - 4\,n x^{3} + 18\,n^{2} x^{2} + 8\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 32\,x^{2} - 40\,x^{3} - 16\,x^{4} - 12\,n x^{2} - 36\,n x^{3} - 24\,n x^{4} - 4\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	16							20
3	8	32							40
4	16	128	192						336
5	32	256	384						672
6	64	768	2304	2048					5184
7	128	1536	4608	4096					10368
8	256	4096	18432	32768	20480				76032
9	512	8192	36864	65536	40960				152064
10	1024	20480	122880	327680	409600	196608			1078272
11	2048	40960	245760	655360	819200	393216			2156544
12	4096	98304	737280	2621440	4915200	4718592	1835008		14929920
13	8192	196608	1474560	5242880	9830400	9437184	3670016		29859840
14	16384	458752	4128768	18350080	45875200	66060288	51380224	16777216	203046912

Technical details

Node id: n-6cfb5fe6d2148054 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-4a9146a6908256f6

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:   4,   16;
  3:   8,   32;
  4:  16,  128,   192;
  5:  32,  256,   384;
  6:  64,  768,  2304,  2048;
  7: 128, 1536,  4608,  4096;
  8: 256, 4096, 18432, 32768, 20480;
  9: 512, 8192, 36864, 65536, 40960;

%F Formula

The row polynomials satisfy (2+(2*n+3)*x+(n^2+n)*x^2)*P(n,x) = (4+(20*n-10)*x+(18*n^2+10*n-24)*x^2+(4*n^3+8*n^2-4*n-8)*x^3)*P(n-1,x) - (8*x+(12*n+32)*x^2+(4*n^2+36*n+40)*x^3+(8*n^2+24*n+16)*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[((4 + (20*n - 10)*x + (18*n^2 + 10*n - 24)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*P[n - 1, x] - (8*x + (12*n + 32)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (8*n^2 + 24*n + 16)*x^4)*D[P[n - 1, x], x])/(2 + (2*n + 3)*x + (n^2 + n)*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] = ((4 + (20*n - 10)*x + (18*n^2 + 10*n - 24)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*v[i - 1] - (8*x + (12*n + 32)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (8*n^2 + 24*n + 16)*x^4)*deriv(v[i - 1], x))/(2 + (2*n + 3)*x + (n^2 + n)*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.