A144443 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 6\,n\right)\,P_{n}(x) = \left(-36\,x^{2} - 216\,n x^{2}\right)\,P_{n-4}(x) + \left(-12\,x - 12\,x^{2} - 72\,n x + 1152\,n x^{2} - 432\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-36\,x^{2} + 36\,x^{3} - 216\,n x^{2} + 216\,n x^{3}\right)\,P_{n-3}'(x) + \left(-7 - 26\,x + 425\,x^{2} - 42\,n + 264\,n x - 1170\,n x^{2} - 72\,n^{2} x + 972\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 144\,x^{2} + 150\,x^{3} - 36\,n x + 396\,n x^{2} - 360\,n x^{3} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-34 + 110\,x + 48\,n - 192\,n x + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 30\,x^{2} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 20 + 2\,x\)

\(P_{3}(x) = 159 + 318\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	20	2						22
3	159	318	3					480
4	1138	8508	3414	4				13064
5	7997	155756	233634	31988	5			429380
6	56016	2438334	8954664	4876668	280080	6		16605768
7	392155	35267298	262977945	350637260	88168245	2352930	7	739795840

All 15 rows

Technical details

Node id: n-f4d3223bc24595a5 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ae7e1082727f92c8

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:        0;
  1:        1;
  2:       20,          2;
  3:      159,        318,            3;
  4:     1138,       8508,         3414,            4;
  5:     7997,     155756,       233634,        31988,            5;
  6:    56016,    2438334,      8954664,      4876668,       280080,            6;
  7:   392155,   35267298,    262977945,    350637260,     88168245,      2352930,           7;
  8:  2745134,  488166536,   6658455318,  17603500312,  11097425530,   1464499608,    19215938,         8;
  9: 19215993, 6586952352, 153921299472, 717407484504, 896759355630, 307842598944, 23054333232, 153727944, 9;

%F Formula

The row polynomials satisfy (6*n-5)*P(n,x) = -(216*n+36)*x^2*P(n-4,x) - ((72*n+12)*x+(432*n^2-1152*n+12)*x^2)*P(n-3,x) - ((216*n+36)*x^2-(216*n+36)*x^3)*d/dx P(n-3,x) - (42*n+7+(72*n^2-264*n+26)*x+(216*n^3-972*n^2+1170*n-425)*x^2)*P(n-2,x) - ((36*n+6)*x+(216*n^2-396*n+144)*x^2-(216*n^2-360*n+150)*x^3)*d/dx P(n-2,x) + (48*n-34+(72*n^2-192*n+110)*x)*P(n-1,x) + ((36*n-30)*x-(36*n-30)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 20 + 2*x, P(3,x) = 159 + 318*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 20 + 2*x;
P[3, x_] := 159 + 318*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(216*n + 36)*x^2*P[n - 4, x] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*P[n - 3, x] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*D[P[n - 3, x], x] - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*P[n - 2, x] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*D[P[n - 2, x], x] + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*P[n - 1, x] + ((36*n - 30)*x - (36*n - 30)*x^2)*D[P[n - 1, x], x])/(6*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 20 + 2*x; v[4] = 159 + 318*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(216*n + 36)*x^2*v[i - 4] - ((72*n + 12)*x + (432*n^2 - 1152*n + 12)*x^2)*v[i - 3] - ((216*n + 36)*x^2 - (216*n + 36)*x^3)*deriv(v[i - 3], x) - (42*n + 7 + (72*n^2 - 264*n + 26)*x + (216*n^3 - 972*n^2 + 1170*n - 425)*x^2)*v[i - 2] - ((36*n + 6)*x + (216*n^2 - 396*n + 144)*x^2 - (216*n^2 - 360*n + 150)*x^3)*deriv(v[i - 2], x) + (48*n - 34 + (72*n^2 - 192*n + 110)*x)*v[i - 1] + ((36*n - 30)*x - (36*n - 30)*x^2)*deriv(v[i - 1], x))/(6*n - 5)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (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.