A156366 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(-10 - 7\,x + 6\,x^{2} - 45\,x^{3} - 54\,x^{4} - 6\,n - 27\,n x - 66\,n x^{2} - 108\,n x^{3} - 54\,n x^{4} - 18\,n^{2} x - 63\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - x^{2} + 21\,x^{3} + 63\,x^{4} + 54\,x^{5} - 3\,n x - 12\,n x^{2} + 36\,n x^{3} + 81\,n x^{4} - 9\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(9 + 28\,x + 27\,x^{2} + 18\,x^{3} + 9\,n + 27\,n x + 45\,n x^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 18\,x^{3} + 3\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 13 + 30\,x + 9\,x^{2}\)

\(P_{1}(x) = 34 + 231\,x + 180\,x^{2} + 27\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	13	30	9								52
1	34	231	180	27							472
2	79	1266	2700	1026	81						5152
3	172	5607	27180	26622	5832	243					65656
4	361	21798	206415	451116	242271	33534	729				956224
5	742	78015	1303776	5482269	6481134	2123577	195372	2187			15667072
6	1507	264450	7278426	52987878	119855700	85336254	18334350	1150362	6561		285215488
7	3040	864159	37301112	436844880	1698218784	2309425830	1064217528	157796424	6823440	19683	5711514880

All 15 rows

Technical details

Node id: n-aca4ba97e8bd9633 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e37acfaae6dc284d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:    13,      30,         9;
  1:    34,     231,       180,          27;
  2:    79,    1266,      2700,        1026,           81;
  3:   172,    5607,     27180,       26622,         5832,          243;
  4:   361,   21798,    206415,      451116,       242271,        33534,           729;
  5:   742,   78015,   1303776,     5482269,      6481134,      2123577,        195372,         2187;
  6:  1507,  264450,   7278426,    52987878,    119855700,     85336254,      18334350,      1150362,         6561;
  7:  3040,  864159,  37301112,   436844880,   1698218784,   2309425830,    1064217528,    157796424,      6823440,       19683;
  8:  6109, 2753574, 179856261,  3214643112,  19893748554,  46415502900,   40832282898,  12830639112,   1362122649,    40665078,     59049;
  9: 12250, 8621127, 829506924, 21762253071, 202824223884, 750319224462, 1133740974960, 679769712918, 151565674122, 11824778763, 243045684, 177147;

%F Formula

The row polynomials satisfy (3*n+2+6*x)*P(n,x) = -(6*n+10+(18*n^2+27*n+7)*x+(27*n^3+63*n^2+66*n-6)*x^2+(81*n^2+108*n+45)*x^3+(54*n+54)*x^4)*P(n-2,x) - ((3*n+5)*x+(9*n^2+12*n+1)*x^2-(27*n^2+36*n+21)*x^3-(81*n+63)*x^4-54*x^5)*d/dx P(n-2,x) + (9*n+9+(18*n^2+27*n+28)*x+(45*n+27)*x^2+18*x^3)*P(n-1,x) + ((3*n+2)*x-9*n*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 13 + 30*x + 9*x^2, P(1,x) = 34 + 231*x + 180*x^2 + 27*x^3.

%t Mathematica

P[0, x_] := 13 + 30*x + 9*x^2;
P[1, x_] := 34 + 231*x + 180*x^2 + 27*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 10 + (18*n^2 + 27*n + 7)*x + (27*n^3 + 63*n^2 + 66*n - 6)*x^2 + (81*n^2 + 108*n + 45)*x^3 + (54*n + 54)*x^4)*P[n - 2, x] - ((3*n + 5)*x + (9*n^2 + 12*n + 1)*x^2 - (27*n^2 + 36*n + 21)*x^3 - (81*n + 63)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (9*n + 9 + (18*n^2 + 27*n + 28)*x + (45*n + 27)*x^2 + 18*x^3)*P[n - 1, x] + ((3*n + 2)*x - 9*n*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n + 2 + 6*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 13 + 30*x + 9*x^2; v[2] = 34 + 231*x + 180*x^2 + 27*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n + 10 + (18*n^2 + 27*n + 7)*x + (27*n^3 + 63*n^2 + 66*n - 6)*x^2 + (81*n^2 + 108*n + 45)*x^3 + (54*n + 54)*x^4)*v[i - 2] - ((3*n + 5)*x + (9*n^2 + 12*n + 1)*x^2 - (27*n^2 + 36*n + 21)*x^3 - (81*n + 63)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (9*n + 9 + (18*n^2 + 27*n + 28)*x + (45*n + 27)*x^2 + 18*x^3)*v[i - 1] + ((3*n + 2)*x - 9*n*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n + 2 + 6*x)); 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 of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.