A152198 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 7\,x + x^{2} - 6\,x^{3} + 10\,n x + 6\,n x^{2} - 3\,n x^{3} + 14\,n^{2} x^{2} + 11\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-50\,x - 110\,x^{2} - 108\,x^{3} - 40\,x^{4} - 16\,x^{5} + 4\,n + 50\,n x - 38\,n x^{2} - 136\,n x^{3} - 124\,n x^{4} - 32\,n x^{5} + 17\,n^{2} x + 149\,n^{2} x^{2} + 109\,n^{2} x^{3} + 16\,n^{2} x^{4} - 4\,n^{2} x^{5} + 20\,n^{3} x^{2} + 135\,n^{3} x^{3} + 109\,n^{3} x^{4} + 28\,n^{3} x^{5} + 8\,n^{4} x^{3} + 42\,n^{4} x^{4} + 20\,n^{4} x^{5} + n^{5} x^{4} + 4\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1 - 17\,x - 69\,x^{2} - 132\,x^{3} - 148\,x^{4} - 96\,x^{5} - 32\,x^{6} - 6\,n x - 79\,n x^{2} - 273\,n x^{3} - 410\,n x^{4} - 312\,n x^{5} - 96\,n x^{6} - 11\,n^{2} x^{2} - 107\,n^{2} x^{3} - 287\,n^{2} x^{4} - 286\,n^{2} x^{5} - 104\,n^{2} x^{6} - 6\,n^{3} x^{3} - 46\,n^{3} x^{4} - 92\,n^{3} x^{5} - 48\,n^{3} x^{6} - n^{4} x^{4} - 6\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	2	16			18
3	2	16			18
4	4	64	144		212
5	4	64	144		212
6	8	192	864	1024	2088
7	8	192	864	1024	2088

All 15 rows

Technical details

Node id: n-060edc9e431477ca · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-5367d60e6d4311ec

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:  1;
  2:  2,  16;
  3:  2,  16;
  4:  4,  64,  144;
  5:  4,  64,  144;
  6:  8, 192,  864, 1024;
  7:  8, 192,  864, 1024;
  8: 16, 512, 3456, 8192, 6400;
  9: 16, 512, 3456, 8192, 6400;

%F Formula

The row polynomials satisfy (4+(10*n+7)*x+(14*n^2+6*n+1)*x^2+(6*n^3+11*n^2-3*n-6)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (4*n+(17*n^2+50*n-50)*x+(20*n^3+149*n^2-38*n-110)*x^2+(8*n^4+135*n^3+109*n^2-136*n-108)*x^3+(n^5+42*n^4+109*n^3+16*n^2-124*n-40)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (1+(6*n+17)*x+(11*n^2+79*n+69)*x^2+(6*n^3+107*n^2+273*n+132)*x^3+(n^4+46*n^3+287*n^2+410*n+148)*x^4+(6*n^4+92*n^3+286*n^2+312*n+96)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*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[((4*n + (17*n^2 + 50*n - 50)*x + (20*n^3 + 149*n^2 - 38*n - 110)*x^2 + (8*n^4 + 135*n^3 + 109*n^2 - 136*n - 108)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (1 + (6*n + 17)*x + (11*n^2 + 79*n + 69)*x^2 + (6*n^3 + 107*n^2 + 273*n + 132)*x^3 + (n^4 + 46*n^3 + 287*n^2 + 410*n + 148)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*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] = ((4*n + (17*n^2 + 50*n - 50)*x + (20*n^3 + 149*n^2 - 38*n - 110)*x^2 + (8*n^4 + 135*n^3 + 109*n^2 - 136*n - 108)*x^3 + (n^5 + 42*n^4 + 109*n^3 + 16*n^2 - 124*n - 40)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*v[i - 1] - (1 + (6*n + 17)*x + (11*n^2 + 79*n + 69)*x^2 + (6*n^3 + 107*n^2 + 273*n + 132)*x^3 + (n^4 + 46*n^3 + 287*n^2 + 410*n + 148)*x^4 + (6*n^4 + 92*n^3 + 286*n^2 + 312*n + 96)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*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 basis_map_A013611 of Triangle read by rows, A007318 rows repeated.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, A007318 rows repeated.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    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
    
    Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    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
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
    
    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.