A038259 → 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

\(P_{n}(x) = \left(11 + 5\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	11	5							16
2	121	110	25						256
3	1331	1815	825	125					4096
4	14641	26620	18150	5500	625				65536
5	161051	366025	332750	151250	34375	3125			1048576
6	1771561	4831530	5490375	3327500	1134375	206250	15625		16777216
7	19487171	62004635	84551775	64054375	29115625	7940625	1203125	78125	268435456

Show all 15 stored rows

Generated b-files: A001025 (row sums), A038253 (reversed triangle), A038319 ?

Computed in your browser ?

Technical details

Node id: n-0195873a37fb074a · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-2235b7dd30e0ad6a

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 ?

A001025: 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 ?
A038319: triangle, 36 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A038253: reversed triangle, 33 terms agree, 7 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 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:         11,          5;
  2:        121,        110,          25;
  3:       1331,       1815,         825,         125;
  4:      14641,      26620,       18150,        5500,         625;
  5:     161051,     366025,      332750,      151250,       34375,       3125;
  6:    1771561,    4831530,     5490375,     3327500,     1134375,     206250,      15625;
  7:   19487171,   62004635,    84551775,    64054375,    29115625,    7940625,    1203125,     78125;
  8:  214358881,  779486840,  1240092700,  1127357000,   640543750,  232925000,   52937500,   6875000,   390625;
  9: 2357947691, 9646149645, 17538453900, 18601390500, 12682766250, 5764893750, 1746937500, 340312500, 38671875, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (11+5*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[(11 + 5*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] = (11 + 5*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.

A038253: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:       1;
  1:       5,       11;
  2:      25,      110,       121;
  3:     125,      825,      1815,       1331;
  4:     625,     5500,     18150,      26620,      14641;
  5:    3125,    34375,    151250,     332750,     366025,      161051;
  6:   15625,   206250,   1134375,    3327500,    5490375,     4831530,     1771561;
  7:   78125,  1203125,   7940625,   29115625,   64054375,    84551775,    62004635,    19487171;
  8:  390625,  6875000,  52937500,  232925000,  640543750,  1127357000,  1240092700,   779486840,  214358881;
  9: 1953125, 38671875, 340312500, 1746937500, 5764893750, 12682766250, 18601390500, 17538453900, 9646149645, 2357947691;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+5*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[(11 + 5*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] = (11 + 5*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.

A038319: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:          1;
  1:         11,          5;
  2:        121,        110,          25;
  3:       1331,       1815,         825,         125;
  4:      14641,      26620,       18150,        5500,         625;
  5:     161051,     366025,      332750,      151250,       34375,       3125;
  6:    1771561,    4831530,     5490375,     3327500,     1134375,     206250,      15625;
  7:   19487171,   62004635,    84551775,    64054375,    29115625,    7940625,    1203125,     78125;
  8:  214358881,  779486840,  1240092700,  1127357000,   640543750,  232925000,   52937500,   6875000,   390625;
  9: 2357947691, 9646149645, 17538453900, 18601390500, 12682766250, 5764893750, 1746937500, 340312500, 38671875, 1953125;

%F Formula

The row polynomials satisfy P(n,x) = (11+5*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[(11 + 5*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] = (11 + 5*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-1194f7dc6f45a71b
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-1194f7dc6f45a71b
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-1194f7dc6f45a71b

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-1194f7dc6f45a71b

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-0a7984b7b9125974 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-7949976749bfedcf root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

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