A146987 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 72\,x + 36\,x^{2} - 12\,n - 24\,n x - 12\,n x^{2}\right)\,P_{n-2}(x) + \left(-24 - 24\,x + 8\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 7 + 14\,x + 4\,x^{2}\)

\(P_{3}(x) = 26 + 78\,x + 60\,x^{2} + 8\,x^{3}\)

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	2														4
2	7	14	4													25
3	26	78	60	8												172
4	124	496	636	280	16											1552
5	680	3400	6152	4856	1456	32										16576
6	3952	23712	55392	63488	35952	8160	64									190720
7	23456	164192	469248	704320	587680	259296	47552	128								2255872
8	140224	1121792	3786304	7012736	7716160	5053184	1826752	281984	256							26939392
9	840320	7562880	29411712	64708224	88244352	76487040	41193600	12615552	1684224	512						322748416
10	5039872	50398720	221755392	564473856	917285376	987872256	705653760	322609152	85706496	10087936	1024					3870883840
11	30235136	332586496	1632699392	4716699648	8889203712	11429053440	10159263744	6168225792	2449218048	574541312	60488704	2048				46442217472
12	181402624	2176831488	11791174656	38094592000	81631365120	121903054848	129522333696	97958449152	51700608000	18140753920	3809639424	362846208	4096			557273055232
13	1088399360	14149191680	83806758912	299309914112	718344028160	1221185433600	1508524216320	1364856582144	897933256704	419036456960	131697674240	25033635840	2176888832	8192		6687142445056
14	6530363392	91425087488	587732721664	2298688110592	6105890852864	11637111169024	16378159497216	17240172331008	13576640630784	7901752680448	3304371986432	940375932928	163260166144	13060923392	16384	80245172469760

Technical details

Node id: n-9addcdd37c167f65 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-48e472502fc99343

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,       2;
  2:      7,      14,        4;
  3:     26,      78,       60,        8;
  4:    124,     496,      636,      280,       16;
  5:    680,    3400,     6152,     4856,     1456,       32;
  6:   3952,   23712,    55392,    63488,    35952,     8160,       64;
  7:  23456,  164192,   469248,   704320,   587680,   259296,    47552,      128;
  8: 140224, 1121792,  3786304,  7012736,  7716160,  5053184,  1826752,   281984,     256;
  9: 840320, 7562880, 29411712, 64708224, 88244352, 76487040, 41193600, 12615552, 1684224, 512;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(24*n-72)*x+(12*n-36)*x^2)*P(n-2,x) + (8*n-24+(8*n-24)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 7 + 14*x + 4*x^2, P(3,x) = 26 + 78*x + 60*x^2 + 8*x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 7 + 14*x + 4*x^2;
P[3, x_] := 26 + 78*x + 60*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (24*n - 72)*x + (12*n - 36)*x^2)*P[n - 2, x] + (8*n - 24 + (8*n - 24)*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 + 2*x; v[3] = 7 + 14*x + 4*x^2; v[4] = 26 + 78*x + 60*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (24*n - 72)*x + (12*n - 36)*x^2)*v[i - 2] + (8*n - 24 + (8*n - 24)*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