A153648 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-30 - 90\,x + 15\,n + 45\,n x\right)\,P_{n-2}(x) + \left(-4 - 6\,x + 2\,n + 3\,n x\right)\,P_{n-1}(x)\)

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

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

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

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	1250	7500	12717	4401	162											26030
5	6250	46875	115974	100008	15957	486										285550
6	31250	281250	900828	1186218	532413	56133	1458									2989550
7	156250	1640625	6494391	11793870	9363195	2434860	193185	4374								32080750
8	781250	9375000	44079327	101401443	115474005	57759885	10238805	653913	13122							339776750
9	3906250	52734375	287527644	804196512	1206324414	919808460	306223740	40783176	2184813	39366						3623728750
10	19531250	292968750	1816323318	5975567316	11120413374	11521318662	6124649760	1470792492	156342069	7223661	118098					38505248750
11	97656250	1611328125	11197514421	42401796246	94451237946	124485602976	92797992306	35707350204	6593076729	582380604	23678649	354294				409949968750
12	488281250	8789062500	67657456737	289764194805	751814594460	1205563301550	1169381879730	647479803906	188838997245	28087739415	2121335325	77058945	1062882			4360064768750
13	2441406250	47607421875	402154582914	1922415852456	5697771173595	10784160138690	12949275682620	9514625153832	4025844316323	928116632430	115068195810	7588977480	249245829	3188646		46397321968750
14	12207031250	256347656250	2357501095008	12442343928894	41479397587683	90576740072115	130042108547010	120211459001964	68564740228587	22852837181079	4310254592460	456998516190	26749019853	801944469	9565938	493590495968750

Technical details

Node id: n-3d81a9f39a5d31c9 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-cd3a6646a84a90ae

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:    1250,     7500,     12717,      4401,        162;
  5:    6250,    46875,    115974,    100008,      15957,       486;
  6:   31250,   281250,    900828,   1186218,     532413,     56133,      1458;
  7:  156250,  1640625,   6494391,  11793870,    9363195,   2434860,    193185,     4374;
  8:  781250,  9375000,  44079327, 101401443,  115474005,  57759885,  10238805,   653913,   13122;
  9: 3906250, 52734375, 287527644, 804196512, 1206324414, 919808460, 306223740, 40783176, 2184813, 39366;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(45*n-90)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 10 + 15*x, P(2,x) = 50 + 150*x + 18*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 15*x;
P[2, x_] := 50 + 150*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (45*n - 90)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];
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; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (45*n - 90)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*x)*v[i - 1])/(n - 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 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.?
  - 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

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