A081582 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(13\,n - 69\,n x + 28\,n x^{2} - 20\,n^{2} + 28\,n^{2} x + 7\,n^{3}\right)\,P_{n}(x) = \left(574\,x^{2} - 392\,x^{3} + 6\,n + 211\,n x - 938\,n x^{2} + 392\,n x^{3} - 13\,n^{2} - 350\,n^{2} x + 392\,n^{2} x^{2} + 7\,n^{3} + 98\,n^{3} x\right)\,P_{n-1}(x) + \left(13\,x^{2} + 336\,x^{3} - 784\,x^{4} + 6\,n x + 140\,n x^{2} - 784\,n x^{3} - 7\,n^{2} x - 196\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	8	7			15
3	15	14			29
4	22	119	49		190
5	29	322	147		498
6	36	623	1323	343	2325
7	43	1022	4606	1372	7043

All 15 rows

Technical details

Node id: n-4daeb6d3857fe217 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-40b06077620fcf4f

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:  8,    7;
  3: 15,   14;
  4: 22,  119,    49;
  5: 29,  322,   147;
  6: 36,  623,  1323,   343;
  7: 43, 1022,  4606,  1372;
  8: 50, 1519, 11025, 13034,  2401;
  9: 57, 2114, 21609, 54880, 12005;

%F Formula

The row polynomials satisfy (7*n^3-20*n^2+13*n+(28*n^2-69*n)*x+28*n*x^2)*P(n,x) = (7*n^3-13*n^2+6*n+(98*n^3-350*n^2+211*n)*x+(392*n^2-938*n+574)*x^2+(392*n-392)*x^3)*P(n-1,x) - ((7*n^2-6*n)*x+(196*n^2-140*n-13)*x^2+(784*n-336)*x^3+784*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[((7*n^3 - 13*n^2 + 6*n + (98*n^3 - 350*n^2 + 211*n)*x + (392*n^2 - 938*n + 574)*x^2 + (392*n - 392)*x^3)*P[n - 1, x] - ((7*n^2 - 6*n)*x + (196*n^2 - 140*n - 13)*x^2 + (784*n - 336)*x^3 + 784*x^4)*D[P[n - 1, x], x])/(7*n^3 - 20*n^2 + 13*n + (28*n^2 - 69*n)*x + 28*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] = ((7*n^3 - 13*n^2 + 6*n + (98*n^3 - 350*n^2 + 211*n)*x + (392*n^2 - 938*n + 574)*x^2 + (392*n - 392)*x^3)*v[i - 1] - ((7*n^2 - 6*n)*x + (196*n^2 - 140*n - 13)*x^2 + (784*n - 336)*x^3 + 784*x^4)*deriv(v[i - 1], x))/(7*n^3 - 20*n^2 + 13*n + (28*n^2 - 69*n)*x + 28*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from gamma_polynomial of Pascal-(1,7,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,7,1) array.?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,7,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,7,1) array.?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.