A081581 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(11\,n - 59\,n x + 24\,n x^{2} - 17\,n^{2} + 24\,n^{2} x + 6\,n^{3}\right)\,P_{n}(x) = \left(420\,x^{2} - 288\,x^{3} + 5\,n + 145\,n x - 684\,n x^{2} + 288\,n x^{3} - 11\,n^{2} - 252\,n^{2} x + 288\,n^{2} x^{2} + 6\,n^{3} + 72\,n^{3} x\right)\,P_{n-1}(x) + \left(11\,x^{2} + 240\,x^{3} - 576\,x^{4} + 5\,n x + 96\,n x^{2} - 576\,n x^{3} - 6\,n^{2} x - 144\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	7	6			13
3	13	12			25
4	19	90	36		145
5	25	240	108		373
6	31	462	864	216	1573
7	37	756	2952	864	4609

All 15 rows

Technical details

Node id: n-656e208fe576c17f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-81cabc8974afa743

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:  7,    6;
  3: 13,   12;
  4: 19,   90,    36;
  5: 25,  240,   108;
  6: 31,  462,   864,   216;
  7: 37,  756,  2952,   864;
  8: 43, 1122,  7020,  7344, 1296;
  9: 49, 1560, 13716, 30240, 6480;

%F Formula

The row polynomials satisfy (6*n^3-17*n^2+11*n+(24*n^2-59*n)*x+24*n*x^2)*P(n,x) = (6*n^3-11*n^2+5*n+(72*n^3-252*n^2+145*n)*x+(288*n^2-684*n+420)*x^2+(288*n-288)*x^3)*P(n-1,x) - ((6*n^2-5*n)*x+(144*n^2-96*n-11)*x^2+(576*n-240)*x^3+576*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[((6*n^3 - 11*n^2 + 5*n + (72*n^3 - 252*n^2 + 145*n)*x + (288*n^2 - 684*n + 420)*x^2 + (288*n - 288)*x^3)*P[n - 1, x] - ((6*n^2 - 5*n)*x + (144*n^2 - 96*n - 11)*x^2 + (576*n - 240)*x^3 + 576*x^4)*D[P[n - 1, x], x])/(6*n^3 - 17*n^2 + 11*n + (24*n^2 - 59*n)*x + 24*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] = ((6*n^3 - 11*n^2 + 5*n + (72*n^3 - 252*n^2 + 145*n)*x + (288*n^2 - 684*n + 420)*x^2 + (288*n - 288)*x^3)*v[i - 1] - ((6*n^2 - 5*n)*x + (144*n^2 - 96*n - 11)*x^2 + (576*n - 240)*x^3 + 576*x^4)*deriv(v[i - 1], x))/(6*n^3 - 17*n^2 + 11*n + (24*n^2 - 59*n)*x + 24*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-(1,6,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,6,1) 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-(1,6,1) 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-(1,6,1) 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.