A274496 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1024 + 448\,x + 16\,x^{2} - 24\,x^{3} + 640\,n x + 96\,n x^{2} - 12\,n x^{3} + 224\,n^{2} x^{2} + 44\,n^{2} x^{3} + n^{2} x^{4} + 24\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(2048 - 3200\,x - 1760\,x^{2} - 464\,x^{3} - 40\,x^{4} - 4\,x^{5} + 5376\,n x - 576\,n x^{2} - 552\,n x^{3} - 124\,n x^{4} - 8\,n x^{5} + 3008\,n^{2} x^{2} + 504\,n^{2} x^{3} + 18\,n^{2} x^{4} - n^{2} x^{5} + 576\,n^{3} x^{3} + 112\,n^{3} x^{4} + 7\,n^{3} x^{5} + 42\,n^{4} x^{4} + 5\,n^{4} x^{5} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-4096\,x - 4608\,x^{2} - 2176\,x^{3} - 608\,x^{4} - 96\,x^{5} - 8\,x^{6} - 4352\,n x^{2} - 4416\,n x^{3} - 1648\,n x^{4} - 312\,n x^{5} - 24\,n x^{6} - 1280\,n^{2} x^{3} - 1104\,n^{2} x^{4} - 284\,n^{2} x^{5} - 26\,n^{2} x^{6} - 128\,n^{3} x^{4} - 88\,n^{3} x^{5} - 12\,n^{3} x^{6} - 4\,n^{4} x^{5} - 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	2				2
2	4	8			12
3	8	16			24
4	16	64	36		116
5	32	128	72		232
6	64	384	432	128	1008
7	128	768	864	256	2016

All 15 rows

Technical details

Node id: n-0a0b20e3f10ebdec · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-f613749a36a48642

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   2;
  2:   4,    8;
  3:   8,   16;
  4:  16,   64,   36;
  5:  32,  128,   72;
  6:  64,  384,  432,  128;
  7: 128,  768,  864,  256;
  8: 256, 2048, 3456, 2048, 400;
  9: 512, 4096, 6912, 4096, 800;

%F Formula

The row polynomials satisfy (1024+(640*n+448)*x+(224*n^2+96*n+16)*x^2+(24*n^3+44*n^2-12*n-24)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (2048+(5376*n-3200)*x+(3008*n^2-576*n-1760)*x^2+(576*n^3+504*n^2-552*n-464)*x^3+(42*n^4+112*n^3+18*n^2-124*n-40)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) - (4096*x+(4352*n+4608)*x^2+(1280*n^2+4416*n+2176)*x^3+(128*n^3+1104*n^2+1648*n+608)*x^4+(4*n^4+88*n^3+284*n^2+312*n+96)*x^5+(2*n^4+12*n^3+26*n^2+24*n+8)*x^6)*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[((2048 + (5376*n - 3200)*x + (3008*n^2 - 576*n - 1760)*x^2 + (576*n^3 + 504*n^2 - 552*n - 464)*x^3 + (42*n^4 + 112*n^3 + 18*n^2 - 124*n - 40)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] - (4096*x + (4352*n + 4608)*x^2 + (1280*n^2 + 4416*n + 2176)*x^3 + (128*n^3 + 1104*n^2 + 1648*n + 608)*x^4 + (4*n^4 + 88*n^3 + 284*n^2 + 312*n + 96)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(1024 + (640*n + 448)*x + (224*n^2 + 96*n + 16)*x^2 + (24*n^3 + 44*n^2 - 12*n - 24)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];
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] = ((2048 + (5376*n - 3200)*x + (3008*n^2 - 576*n - 1760)*x^2 + (576*n^3 + 504*n^2 - 552*n - 464)*x^3 + (42*n^4 + 112*n^3 + 18*n^2 - 124*n - 40)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] - (4096*x + (4352*n + 4608)*x^2 + (1280*n^2 + 4416*n + 2176)*x^3 + (128*n^3 + 1104*n^2 + 1648*n + 608)*x^4 + (4*n^4 + 88*n^3 + 284*n^2 + 312*n + 96)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(1024 + (640*n + 448)*x + (224*n^2 + 96*n + 16)*x^2 + (24*n^3 + 44*n^2 - 12*n - 24)*x^3 + (n^4 + 2*n^3 + n^2)*x^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 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)²

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.