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	x⁴	x⁵	x⁶	x⁷	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
8	50	1519	11025	13034	2401				28029
9	57	2114	21609	54880	12005				90665
10	64	2807	37387	156065	120050	16807			333180
11	71	3598	59388	355348	588245	100842			1107492
12	78	4487	88641	701092	1932805	1058841	117649		3903593
13	85	5474	126175	1251264	5008486	5882450	823543		13097477
14	92	6559	173019	2073435	11092620	21882714	9058973	823543	45110955

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.