A153518 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 45\,x + 5\,n + 15\,n x\right)\,P_{n-2}(x) + \left(-6 - 9\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 15\,x\)

\(P_{2}(x) = 50 + 150\,x + 18\,x^{2}\)

\(P_{3}(x) = 250 + 1125\,x + 1161\,x^{2} + 54\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	15														25
2	50	150	18													218
3	250	1125	1161	54												2590
4	750	4500	8037	3861	162											17310
5	2750	20625	52254	49518	12717	486										138350
6	9250	83250	274068	395658	232713	41553	1458									1037950
7	32250	338625	1368531	2644920	2458755	974430	134865	4374								7956750
8	110750	1329000	6372027	15484743	19950705	13023585	3823605	435213	13122							60542750
9	382750	5167125	28653084	83838132	138323214	127652760	62008740	14386086	1397493	39366						461848750
10	1319250	19788750	124602678	426639636	860185494	1034653662	721447560	274328532	52547049	4468041	118098					3520098750
11	4552250	75112125	529343901	2076074226	4949477946	7362975816	6771691206	3715061274	1150858449	187736454	14230809	354294				26837468750
12	15700750	282613500	2203868817	9742418505	26827700580	47650436190	54759352590	39938552226	17824563945	4638594195	659282085	45172485	1062882			204589318750
13	54162750	1056173625	9028979514	44416973106	138771159795	286841022390	396773107020	364305836682	216944995923	80952439230	18121547610	2283424830	142957629	3188646		1559695968750
14	186829250	3923414250	36485026488	197691009534	691068019383	1630663213815	2642622587010	2940014044404	2215008605007	1103301296379	351975736260	69046586190	7819091433	451193409	9565938	11890266218750

Technical details

Node id: n-361d6b7dd38ebd24 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bdfdb99287736e94

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:      2;
  1:     10,      15;
  2:     50,     150,       18;
  3:    250,    1125,     1161,       54;
  4:    750,    4500,     8037,     3861,       162;
  5:   2750,   20625,    52254,    49518,     12717,       486;
  6:   9250,   83250,   274068,   395658,    232713,     41553,     1458;
  7:  32250,  338625,  1368531,  2644920,   2458755,    974430,   134865,     4374;
  8: 110750, 1329000,  6372027, 15484743,  19950705,  13023585,  3823605,   435213,   13122;
  9: 382750, 5167125, 28653084, 83838132, 138323214, 127652760, 62008740, 14386086, 1397493, 39366;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(15*n-45)*x)*P(n-2,x) + (2*n-6+(3*n-9)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 15*x, P(2,x) = 50 + 150*x + 18*x^2, P(3,x) = 250 + 1125*x + 1161*x^2 + 54*x^3.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 15*x;
P[2, x_] := 50 + 150*x + 18*x^2;
P[3, x_] := 250 + 1125*x + 1161*x^2 + 54*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (15*n - 45)*x)*P[n - 2, x] + (2*n - 6 + (3*n - 9)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 15*x; v[3] = 50 + 150*x + 18*x^2; v[4] = 250 + 1125*x + 1161*x^2 + 54*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (15*n - 45)*x)*v[i - 2] + (2*n - 6 + (3*n - 9)*x)*v[i - 1])/(n - 3)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 3
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.?
  - 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

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