A038327 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(13 + x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	13	1							14
2	169	26	1						196
3	2197	507	39	1					2744
4	28561	8788	1014	52	1				38416
5	371293	142805	21970	1690	65	1			537824
6	4826809	2227758	428415	43940	2535	78	1		7529536
7	62748517	33787663	7797153	999635	76895	3549	91	1	105413504

Show all 15 stored rows

Generated b-files: A001023 (row sums), A123187 (reversed triangle), A133371 ?

Computed in your browser ?

Technical details

Node id: n-7a809bc95dc4396c · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-96a9640a0a811b71

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 ?

A123187: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
A133371: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
A001023: 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:          13,          1;
  2:         169,         26,          1;
  3:        2197,        507,         39,         1;
  4:       28561,       8788,       1014,        52,        1;
  5:      371293,     142805,      21970,      1690,       65,       1;
  6:     4826809,    2227758,     428415,     43940,     2535,      78,      1;
  7:    62748517,   33787663,    7797153,    999635,    76895,    3549,     91,    1;
  8:   815730721,  501988136,  135150652,  20792408,  1999270,  123032,   4732,  104,   1;
  9: 10604499373, 7341576489, 2258946612, 405451956, 46782918, 3598686, 184548, 6084, 117, 1;

%F Formula

The row polynomials satisfy P(n,x) = (13+x)*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[(13 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (13 + 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.

A123187: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  13;
  2: 1,  26,  169;
  3: 1,  39,  507,   2197;
  4: 1,  52, 1014,   8788,   28561;
  5: 1,  65, 1690,  21970,  142805,   371293;
  6: 1,  78, 2535,  43940,  428415,  2227758,   4826809;
  7: 1,  91, 3549,  76895,  999635,  7797153,  33787663,   62748517;
  8: 1, 104, 4732, 123032, 1999270, 20792408, 135150652,  501988136,  815730721;
  9: 1, 117, 6084, 184548, 3598686, 46782918, 405451956, 2258946612, 7341576489, 10604499373;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (13+x)*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[(13 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (13 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A133371: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:           1;
  1:          13,          1;
  2:         169,         26,          1;
  3:        2197,        507,         39,         1;
  4:       28561,       8788,       1014,        52,        1;
  5:      371293,     142805,      21970,      1690,       65,       1;
  6:     4826809,    2227758,     428415,     43940,     2535,      78,      1;
  7:    62748517,   33787663,    7797153,    999635,    76895,    3549,     91,    1;
  8:   815730721,  501988136,  135150652,  20792408,  1999270,  123032,   4732,  104,   1;
  9: 10604499373, 7341576489, 2258946612, 405451956, 46782918, 3598686, 184548, 6084, 117, 1;

%F Formula

The row polynomials satisfy P(n,x) = (13+x)*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[(13 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (13 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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-d06691736a5246b1
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-d06691736a5246b1
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-d06691736a5246b1

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 A007318 e-d06691736a5246b1

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-cf37a3b130851ac7 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-75dac0c4b2fbfacb 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