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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	6109	2753574	179856261	3214643112	19893748554	46415502900	40832282898	12830639112	1362122649	40665078	59049							124772279296
9	12250	8621127	829506924	21762253071	202824223884	750319224462	1133740974960	679769712918	151565674122	11824778763	243045684	177147						2952888205312
10	24535	26659602	3701526984	138372620490	1863615936555	10317468371940	24708160056888	25509995447076	10843264218285	1769998302090	103312130400	1455085458	531441					75259370911744
11	49108	81720759	16112109492	838601555922	15812875494720	125466837102117	448601722163496	735290143266348	539841058718868	167779186255665	20557445697252	908161455906	8719883928	1594323				2055120947067904
12	98257	248943558	68819786523	4896256819428	126092393288721	1386984166189218	7078373315920467	17322879659207832	20249651062272987	10904655289755114	2540773930654233	238401158214852	8025857588547	52284228462	4782969			59860854447751168
13	196558	754985967	289716300360	27760615591761	957186397055754	14219585492205375	100052523986973228	348979531250345625	610073566311019554	524964283796040741	212562369391674384	37905615279269163	2767458977488710	71241594649389	313590579516	14348907		1852581727168724992
14	393163	2282451522	1206048384630	153768105328590	6984851363829990	137226727945537938	1295387320744204182	6212187407112376590	15481305463119485940	19974442473451267830	12974617582296458706	4031513341055083674	559869840873818010	32205647115599670	634596913318050	1881170405514	43046721	60706532109640990720

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.