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	x⁴	x⁵	x⁶	x⁷	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
8	43	1122	7020	7344	1296				16825
9	49	1560	13716	30240	6480				52045
10	55	2070	23688	85320	58320	7776			177229
11	61	2652	37584	193536	278640	46656			559129
12	67	3306	56052	381024	907200	443232	46656		1837537
13	73	4032	79740	679104	2340576	2395008	326592		5825125
14	79	4830	109296	1124280	5171040	8817984	3265920	279936	18773365

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.