A274496 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(32 + 12\,x + 8\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(64 - 40\,x - 24\,x^{2} - 2\,x^{3} + 80\,n x + 10\,n x^{2} - n x^{3} + 18\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-128\,x - 128\,x^{2} - 40\,x^{3} - 4\,x^{4} - 48\,n x^{2} - 36\,n x^{3} - 6\,n x^{4} - 4\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2								2
2	4	4							8
3	8	8							16
4	16	32	12						60
5	32	64	24						120
6	64	192	144	32					432
7	128	384	288	64					864
8	256	1024	1152	512	80				3024
9	512	2048	2304	1024	160				6048
10	1024	5120	7680	5120	1600	192			20736
11	2048	10240	15360	10240	3200	384			41472
12	4096	24576	46080	40960	19200	4608	448		139968
13	8192	49152	92160	81920	38400	9216	896		279936
14	16384	114688	258048	286720	179200	64512	12544	1024	933120

Technical details

Node id: n-0b5fdf5ee862725f · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-90678fce5eba4992

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,    4;
  3:   8,    8;
  4:  16,   32,   12;
  5:  32,   64,   24;
  6:  64,  192,  144,   32;
  7: 128,  384,  288,   64;
  8: 256, 1024, 1152,  512,  80;
  9: 512, 2048, 2304, 1024, 160;

%F Formula

The row polynomials satisfy (32+(8*n+12)*x+(n^2+n)*x^2)*P(n,x) = (64+(80*n-40)*x+(18*n^2+10*n-24)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (128*x+(48*n+128)*x^2+(4*n^2+36*n+40)*x^3+(2*n^2+6*n+4)*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[((64 + (80*n - 40)*x + (18*n^2 + 10*n - 24)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (128*x + (48*n + 128)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(32 + (8*n + 12)*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] = ((64 + (80*n - 40)*x + (18*n^2 + 10*n - 24)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (128*x + (48*n + 128)*x^2 + (4*n^2 + 36*n + 40)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(32 + (8*n + 12)*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 shift_x_plus_one of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry …?
  - Affine, reversal and Möbius substitutions (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
  - 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 (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A152198 → x ↦ x+1 → θ+1

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.