A391533 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

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

\(P_{0}(x) = 9 + 24\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	24	16															49
1	18	84	128	64														294
2	36	240	592	640	256													1764
3	72	624	2144	3648	3072	1024												10584
4	144	1536	6784	15872	20736	14336	4096											63504
5	288	3648	19712	58880	104960	111616	65536	16384										381024
6	576	8448	54016	196608	445440	643072	577536	294912	65536									2286144
7	1152	19200	141824	609280	1677312	3067904	3727360	2899968	1310720	262144								13716864
8	2304	43008	360448	1785856	5791744	12845056	19726336	20709376	14221312	5767168	1048576							82301184
9	4608	95232	892928	5013504	18726912	48857088	90832896	120324096	111280128	68419584	25165824	4194304						493807104
10	9216	208896	2166784	13598720	57507840	172621824	377094144	603979776	703856640	581959680	324009984	109051904	16777216					2962842624
11	18432	454656	5169152	35864576	169410560	575275008	1444675584	2716336128	3823632384	3979345920	2975858688	1514143744	469762048	67108864				17777055744
12	36864	983040	12156928	92405760	482279424	1828192256	5190451200	11211374592	18512609280	23253221376	21869101056	14931722240	6996099072	2013265920	268435456			106662334464
13	73728	2113536	28246016	233439232	1334181888	5585502208	17693671424	43184553984	81870716928	120556879872	136751087616	117339848704	73719087104	32010928128	8589934592	1073741824		639974006784
14	147456	4521984	64946176	579862528	3602120704	16507731968	57729351680	157143793664	336479649792	568596627456	755729694720	781684047872	616797569024	358898204672	145223581696	36507222016	4294967296	3839844040704

Computed in your browser ?

Technical details

Node id: n-d021e887e4fd1eea · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-19b2ced286a4e317

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    9,    24,     16;
  1:   18,    84,    128,      64;
  2:   36,   240,    592,     640,      256;
  3:   72,   624,   2144,    3648,     3072,     1024;
  4:  144,  1536,   6784,   15872,    20736,    14336,     4096;
  5:  288,  3648,  19712,   58880,   104960,   111616,    65536,     16384;
  6:  576,  8448,  54016,  196608,   445440,   643072,   577536,    294912,     65536;
  7: 1152, 19200, 141824,  609280,  1677312,  3067904,  3727360,   2899968,   1310720,   262144;
  8: 2304, 43008, 360448, 1785856,  5791744, 12845056, 19726336,  20709376,  14221312,  5767168,  1048576;
  9: 4608, 95232, 892928, 5013504, 18726912, 48857088, 90832896, 120324096, 111280128, 68419584, 25165824, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 16*x^2.

%t Mathematica

P[0, x_] := 9 + 24*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 24*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 4*x)*v[i - 1]); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-3d28f25fd9699ebd
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-3d28f25fd9699ebd
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-3d28f25fd9699ebd

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A013611 e-3d28f25fd9699ebd

Geometric basis map (elementary) polytool ?
- premise (sequence): The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-3cff49c1888b5e80 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-5e5f630c1e2baa8a root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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