A115951 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-4 - 16\,x + 4\,n + 16\,n x\right)\,P_{n-2}(x) + \left(-2 - 8\,x + 4\,n + 16\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2 + 8\,x\)

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	8														10
2	8	56	96													160
3	32	336	1152	1280												2800
4	136	1888	9696	21760	17920											51400
5	592	10240	70080	236800	394240	258048										970000
6	2624	54336	464640	2097920	5268480	6967296	3784704									18640000
7	11776	284032	2913792	16468480	55336960	110444544	121110528	56229888								362800000
8	53344	1468672	17574144	119311360	502351360	1342365696	2221621248	2080505856	843448320							7128700000
9	243392	7531776	102984192	816291840	4131671040	13841952768	30678810624	43353243648	35424829440	12745441280						141103000000
10	1116928	38375680	590238720	5349857280	31634534400	127467196416	354305064960	670541414400	826579353600	599035740160	193730707456					2809273600000
11	5149696	194512384	3323781120	33907568640	229390909440	1080269955072	3612469690368	8575339069440	14156436602880	15472965713920	10073996787712	2958796259328				56197096000000
12	23835904	981685248	18450736128	209216634880	1593705738240	8589714997248	33580577243136	95918461157376	198620271083520	290698024714240	285365332082688	168651386781696	45368209309696			1128614356000000
13	110690816	4936675328	101212440576	1262823710720	10696125440000	64932531535872	290598633701376	970557612490752	2418513834147840	4439976433418240	5835556369989632	5198605027639296	2812828977201152	697972450918400		22741607080000000
14	515483648	24749656064	549673033728	7483670405120	69767889551360	471062555099136	2375043126263808	9082281142910976	26465677525647360	58470502089359360	96389938571902976	114952193471152128	93730720433831936	46764154211532800	10768717814169600	459548117440000000

Technical details

Node id: n-68250fd6ae5aea2b · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-4db38bf8b0b4386b

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,       8;
  2:      8,      56,        96;
  3:     32,     336,      1152,      1280;
  4:    136,    1888,      9696,     21760,      17920;
  5:    592,   10240,     70080,    236800,     394240,      258048;
  6:   2624,   54336,    464640,   2097920,    5268480,     6967296,     3784704;
  7:  11776,  284032,   2913792,  16468480,   55336960,   110444544,   121110528,    56229888;
  8:  53344, 1468672,  17574144, 119311360,  502351360,  1342365696,  2221621248,  2080505856,   843448320;
  9: 243392, 7531776, 102984192, 816291840, 4131671040, 13841952768, 30678810624, 43353243648, 35424829440, 12745441280;

%F Formula

The row polynomials satisfy n*P(n,x) = (4*n-4+(16*n-16)*x)*P(n-2,x) + (4*n-2+(16*n-8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (16*n - 16)*x)*P[n - 2, x] + (4*n - 2 + (16*n - 8)*x)*P[n - 1, x])/(n)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (16*n - 16)*x)*v[i - 2] + (4*n - 2 + (16*n - 8)*x)*v[i - 1])/(n)); 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 ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - 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