A200536 → 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

\(P_{n}(x) = \left(6 + 21\,x + 18\,x^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	21	18													45
2	36	252	657	756	324											2025
3	216	2268	9882	22869	29646	20412	5832									91125
4	1296	18144	110808	385560	836001	1156680	997272	489888	104976							4100625
5	7776	136080	1069200	4966920	15107310	31436181	45321930	44702280	28868400	11022480	1889568					184528125
6	46656	979776	9412848	54704160	214194780	595275156	1204022961	1785825468	1927753020	1477012320	762440688	238085568	34012224			8303765625
7	279936	6858432	77892192	543530736	2603387304	9054416196	23580422082	46714387797	70741266246	81489745764	70291457208	44025989616	18927802656	4999796928	612220032	373669453125

Show all 15 stored rows

Generated b-files: A009989 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-2ff1c5d05cc7026f · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-10e88990a58171dc

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 ?

A009989: 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:        6,        21,         18;
  2:       36,       252,        657,         756,          324;
  3:      216,      2268,       9882,       22869,        29646,         20412,          5832;
  4:     1296,     18144,     110808,      385560,       836001,       1156680,        997272,         489888,         104976;
  5:     7776,    136080,    1069200,     4966920,     15107310,      31436181,      45321930,       44702280,       28868400,       11022480,         1889568;
  6:    46656,    979776,    9412848,    54704160,    214194780,     595275156,    1204022961,     1785825468,     1927753020,     1477012320,       762440688,       238085568,        34012224;
  7:   279936,   6858432,   77892192,   543530736,   2603387304,    9054416196,   23580422082,    46714387797,    70741266246,    81489745764,     70291457208,     44025989616,     18927802656,      4999796928,      612220032;
  8:  1679616,  47029248,  616419072,  5020372224,  28436528736,  118781183808,  378486244080,   938454682032,  1829897338689,  2815364046096,   3406376196720,   3207091962816,   2303358827616,   1219950450432,   449369503488,   102852965376,    11019960576;
  9: 10077696, 317447424, 4716361728, 43913560320, 287142532416, 1400220906336, 5277179841696, 15717000526416, 37499684748246, 72212212665621, 112499054244738, 141453004737744, 142483855725792, 113417893413216, 69775635377088, 32012985473280, 10314683099136, 2082772548864, 198359290368;

%F Formula

The row polynomials satisfy P(n,x) = (6+21*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[(6 + 21*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] = (6 + 21*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-bf76d3591eaf645b
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-bf76d3591eaf645b
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-bf76d3591eaf645b

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-bf76d3591eaf645b

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-982ad459e6d62a8e 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.

A200536 → basis A007318