A103633 → 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

\(P_{n}(x) = \left(2 + 12\,x + 16\,x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	4														5
2	2	12	16													30
3	2	20	64	64												150
4	4	48	208	384	256											900
5	4	64	400	1216	1792	1024										4500
6	8	144	1056	4032	8448	9216	4096									27000
7	8	176	1632	8256	24576	43008	40960	16384								135000
8	16	384	3968	23040	82176	184320	253952	196608	65536							810000
9	16	448	5504	38912	174336	513024	991232	1212416	851968	262144						4050000
10	32	960	12800	99840	504320	1723392	4034560	6389760	6553600	3932160	1048576					24300000
11	32	1088	16640	151040	903680	3740672	10928128	22528000	32112640	30146560	16777216	4194304				121500000
12	64	2304	37632	368640	2411520	11096064	36818944	88768512	154337280	188743680	154140672	75497472	16777216			729000000
13	64	2560	46848	519168	3886080	20742144	81203200	236044288	509411328	806092800	909115392	692060160	318767104	67108864		3645000000
14	128	5376	103936	1225728	9848832	57028608	245374976	796901376	1962999808	3649830912	5042601984	5020581888	3405774848	1409286144	268435456	21870000000

Computed in your browser ?

Technical details

Node id: n-43577dab61c337fb · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-5b8f1017b01dd220

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): 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:  1;
  1:  1,   4;
  2:  2,  12,   16;
  3:  2,  20,   64,    64;
  4:  4,  48,  208,   384,    256;
  5:  4,  64,  400,  1216,   1792,   1024;
  6:  8, 144, 1056,  4032,   8448,   9216,   4096;
  7:  8, 176, 1632,  8256,  24576,  43008,  40960,   16384;
  8: 16, 384, 3968, 23040,  82176, 184320, 253952,  196608,  65536;
  9: 16, 448, 5504, 38912, 174336, 513024, 991232, 1212416, 851968, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (2+12*x+16*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 12*x + 16*x^2)*P[n - 2, x]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 12*x + 16*x^2)*v[i - 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)

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-226c3d68bb34828a
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-226c3d68bb34828a
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-226c3d68bb34828a

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-226c3d68bb34828a

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A152842 → x ↦ x/(x+1)