A146987 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 144\,x + 144\,x^{2} - 12\,n - 48\,n x - 48\,n x^{2}\right)\,P_{n-2}(x) + \left(-24 - 48\,x + 8\,n + 16\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 7 + 28\,x + 16\,x^{2}\)

\(P_{3}(x) = 26 + 156\,x + 240\,x^{2} + 64\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	7	28	16													51
3	26	156	240	64												486
4	124	992	2544	2240	256											6156
5	680	6800	24608	38848	23296	1024										95256
6	3952	47424	221568	507904	575232	261120	4096									1621296
7	23456	328384	1876992	5634560	9402880	8297472	3043328	16384								28623456
8	140224	2243584	15145216	56101888	123458560	161701888	116912128	36093952	65536							511862976
9	840320	15125760	117646848	517665792	1411909632	2447585280	2636390400	1614790656	431161344	262144						9193378176
10	5039872	100797440	887021568	4515790848	14676566016	31611912192	45161840640	41293971456	21940862976	5165023232	1048576					165359874816
11	30235136	665172992	6530797568	37733597184	142227259392	365729710080	650192879616	789532901376	626999820288	294165151744	61940432896	4194304				2975752152576
12	181402624	4353662976	47164698624	304756736000	1306101841920	3900897755136	8289429356544	12538681491456	13235355648000	9288066007040	3901070770176	743109033984	16777216			53559185181696
13	1088399360	28298383360	335227035648	2394479312896	11493504450560	39077933875200	96545549844480	174701642514432	229870913716224	214546665963520	134858418421760	51268886200320	8916536655872	67108864		964039211882496
14	6530363392	182850174976	2350930886656	18389504884736	97694253645824	372387557408768	1048202207821824	2206742058369024	3475620001480704	4045697372389376	3383676914106368	1925889910636544	668713640525824	106995084427264	268435456	17352549085556736

Technical details

Node id: n-3f2d3a36a2f92f4a · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-28c423639b74b313

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:      1;
  1:      2,        4;
  2:      7,       28,        16;
  3:     26,      156,       240,        64;
  4:    124,      992,      2544,      2240,        256;
  5:    680,     6800,     24608,     38848,      23296,       1024;
  6:   3952,    47424,    221568,    507904,     575232,     261120,       4096;
  7:  23456,   328384,   1876992,   5634560,    9402880,    8297472,    3043328,      16384;
  8: 140224,  2243584,  15145216,  56101888,  123458560,  161701888,  116912128,   36093952,     65536;
  9: 840320, 15125760, 117646848, 517665792, 1411909632, 2447585280, 2636390400, 1614790656, 431161344, 262144;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(48*n-144)*x+(48*n-144)*x^2)*P(n-2,x) + (8*n-24+(16*n-48)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 7 + 28*x + 16*x^2, P(3,x) = 26 + 156*x + 240*x^2 + 64*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 7 + 28*x + 16*x^2;
P[3, x_] := 26 + 156*x + 240*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (48*n - 144)*x + (48*n - 144)*x^2)*P[n - 2, x] + (8*n - 24 + (16*n - 48)*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] = 1; v[2] = 2 + 4*x; v[3] = 7 + 28*x + 16*x^2; v[4] = 26 + 156*x + 240*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (48*n - 144)*x + (48*n - 144)*x^2)*v[i - 2] + (8*n - 24 + (16*n - 48)*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 Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.?
  - 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