A139548 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(8 + 24\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	8	24	18													50
2	64	384	864	864	324											2500
3	512	4608	17280	34560	38880	23328	5832									125000
4	4096	49152	258048	774144	1451520	1741824	1306368	559872	104976							6250000
5	32768	491520	3317760	13271040	34836480	62705664	78382080	67184640	37791360	12597120	1889568					312500000
6	262144	4718592	38928384	194641920	656916480	1576599552	2759049216	3547348992	3325639680	2217093120	997691904	272097792	34012224			15625000000
7	2097152	44040192	429391872	2576351232	10627448832	31882346496	71735279616	122974765056	161404379136	161404379136	121053284352	66029064192	24760899072	5714053632	612220032	781250000000

Show all 15 stored rows

Generated b-files: A165800 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-87b7b64af43d2638 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-6b852c5cd7d4c499

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A165800: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:         8,         24,          18;
  2:        64,        384,         864,          864,           324;
  3:       512,       4608,       17280,        34560,         38880,         23328,           5832;
  4:      4096,      49152,      258048,       774144,       1451520,       1741824,        1306368,         559872,          104976;
  5:     32768,     491520,     3317760,     13271040,      34836480,      62705664,       78382080,       67184640,        37791360,        12597120,         1889568;
  6:    262144,    4718592,    38928384,    194641920,     656916480,    1576599552,     2759049216,     3547348992,      3325639680,      2217093120,       997691904,       272097792,        34012224;
  7:   2097152,   44040192,   429391872,   2576351232,   10627448832,   31882346496,    71735279616,   122974765056,    161404379136,    161404379136,    121053284352,     66029064192,     24760899072,      5714053632,       612220032;
  8:  16777216,  402653184,  4529848320,  31708938240,  154581073920,  556491866112,  1530352631808,  3279327068160,   5533864427520,   7378485903360,   7747410198528,   6338790162432,   3961743851520,   1828497162240,    587731230720,   117546246144,    11019960576;
  9: 134217728, 3623878656, 46204452864, 369635622912, 2079200378880, 8732641591296, 28381085171712, 72979933298688, 150521112428544, 250868520714240, 338672502964224, 369460912324608, 323278298284032, 223808052658176, 119897171066880, 47958868426752, 13488431745024, 2380311484416, 198359290368;

%F Formula

The row polynomials satisfy P(n,x) = (8+24*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x + 18*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8 + 24*x + 18*x^2)*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-a142fb8711dcbce4
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-a142fb8711dcbce4
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-a142fb8711dcbce4

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 A013610 e-a142fb8711dcbce4

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-4755dd2fd7289161 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.

A034870 → basis A013610