A274498 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(324 + 189\,x + 9\,x^{2} - 18\,x^{3} + 270\,n x + 54\,n x^{2} - 9\,n x^{3} + 126\,n^{2} x^{2} + 33\,n^{2} x^{3} + n^{2} x^{4} + 18\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(972 - 2025\,x - 1485\,x^{2} - 522\,x^{3} - 60\,x^{4} - 8\,x^{5} + 3402\,n x - 486\,n x^{2} - 621\,n x^{3} - 186\,n x^{4} - 16\,n x^{5} + 2538\,n^{2} x^{2} + 567\,n^{2} x^{3} + 27\,n^{2} x^{4} - 2\,n^{2} x^{5} + 648\,n^{3} x^{3} + 168\,n^{3} x^{4} + 14\,n^{3} x^{5} + 63\,n^{4} x^{4} + 10\,n^{4} x^{5} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1944\,x - 2916\,x^{2} - 1836\,x^{3} - 684\,x^{4} - 144\,x^{5} - 16\,x^{6} - 2754\,n x^{2} - 3726\,n x^{3} - 1854\,n x^{4} - 468\,n x^{5} - 48\,n x^{6} - 1080\,n^{2} x^{3} - 1242\,n^{2} x^{4} - 426\,n^{2} x^{5} - 52\,n^{2} x^{6} - 144\,n^{3} x^{4} - 132\,n^{3} x^{5} - 24\,n^{3} x^{6} - 6\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	9	24			33
3	27	72			99
4	81	432	324		837
5	243	1296	972		2511
6	729	5832	8748	3456	18765
7	2187	17496	26244	10368	56295

All 15 rows

Technical details

Node id: n-9a8cb397421dd15e · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-4611b6d868b2cab4

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:     3;
  2:     9,     24;
  3:    27,     72;
  4:    81,    432,    324;
  5:   243,   1296,    972;
  6:   729,   5832,   8748,   3456;
  7:  2187,  17496,  26244,  10368;
  8:  6561,  69984, 157464, 124416, 32400;
  9: 19683, 209952, 472392, 373248, 97200;

%F Formula

The row polynomials satisfy (324+(270*n+189)*x+(126*n^2+54*n+9)*x^2+(18*n^3+33*n^2-9*n-18)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (972+(3402*n-2025)*x+(2538*n^2-486*n-1485)*x^2+(648*n^3+567*n^2-621*n-522)*x^3+(63*n^4+168*n^3+27*n^2-186*n-60)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (1944*x+(2754*n+2916)*x^2+(1080*n^2+3726*n+1836)*x^3+(144*n^3+1242*n^2+1854*n+684)*x^4+(6*n^4+132*n^3+426*n^2+468*n+144)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*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[((972 + (3402*n - 2025)*x + (2538*n^2 - 486*n - 1485)*x^2 + (648*n^3 + 567*n^2 - 621*n - 522)*x^3 + (63*n^4 + 168*n^3 + 27*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (1944*x + (2754*n + 2916)*x^2 + (1080*n^2 + 3726*n + 1836)*x^3 + (144*n^3 + 1242*n^2 + 1854*n + 684)*x^4 + (6*n^4 + 132*n^3 + 426*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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] = ((972 + (3402*n - 2025)*x + (2538*n^2 - 486*n - 1485)*x^2 + (648*n^3 + 567*n^2 - 621*n - 522)*x^3 + (63*n^4 + 168*n^3 + 27*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (1944*x + (2754*n + 2916)*x^2 + (1080*n^2 + 3726*n + 1836)*x^3 + (144*n^3 + 1242*n^2 + 1854*n + 684)*x^4 + (6*n^4 + 132*n^3 + 426*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetr…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0…?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

A188440 → basis A007318 → (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.