A038322: Triangle whose (i,j)-th entry is binomial(i,j)11^(i-j)8^j.

A038322 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A038322

\(P_{n}(x) = \left(11 + 8\,x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	11	8							19
2	121	176	64						361
3	1331	2904	2112	512					6859
4	14641	42592	46464	22528	4096				130321
5	161051	585640	851840	619520	225280	32768			2476099
6	1771561	7730448	14055360	13629440	7434240	2162688	262144		47045881
7	19487171	99207416	216452544	262366720	190812160	83263488	20185088	2097152	893871739

Show all 15 stored rows

Generated b-files: A001029 (row sums), A038289 (reversed triangle), A038322 ?

Computed in your browser ?

Technical details

Node id: n-2f02b2035f8f4541 · revision 1 · JSON
Definition source: oeis_catalog:A038322
Fingerprint: f6-e348a50f4e54dede

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 ?

A038322: 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 ?
A001029: 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 ?
A038289: reversed triangle, 32 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,           8;
  2:        121,         176,          64;
  3:       1331,        2904,        2112,         512;
  4:      14641,       42592,       46464,       22528,        4096;
  5:     161051,      585640,      851840,      619520,      225280,       32768;
  6:    1771561,     7730448,    14055360,    13629440,     7434240,     2162688,      262144;
  7:   19487171,    99207416,   216452544,   262366720,   190812160,    83263488,    20185088,    2097152;
  8:  214358881,  1247178944,  3174637312,  4617654272,  4197867520,  2442395648,   888143872,  184549376,   16777216;
  9: 2357947691, 15433839432, 44898441984, 76191295488, 83117776896, 60449292288, 29308747776, 9135194112, 1660944384, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (11+8*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 + 8*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 + 8*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.

A038289: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:         1;
  1:         8,         11;
  2:        64,        176,        121;
  3:       512,       2112,       2904,        1331;
  4:      4096,      22528,      46464,       42592,       14641;
  5:     32768,     225280,     619520,      851840,      585640,      161051;
  6:    262144,    2162688,    7434240,    13629440,    14055360,     7730448,     1771561;
  7:   2097152,   20185088,   83263488,   190812160,   262366720,   216452544,    99207416,    19487171;
  8:  16777216,  184549376,  888143872,  2442395648,  4197867520,  4617654272,  3174637312,  1247178944,   214358881;
  9: 134217728, 1660944384, 9135194112, 29308747776, 60449292288, 83117776896, 76191295488, 44898441984, 15433839432, 2357947691;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+8*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 + 8*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 + 8*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.

A038322: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:          1;
  1:         11,           8;
  2:        121,         176,          64;
  3:       1331,        2904,        2112,         512;
  4:      14641,       42592,       46464,       22528,        4096;
  5:     161051,      585640,      851840,      619520,      225280,       32768;
  6:    1771561,     7730448,    14055360,    13629440,     7434240,     2162688,      262144;
  7:   19487171,    99207416,   216452544,   262366720,   190812160,    83263488,    20185088,    2097152;
  8:  214358881,  1247178944,  3174637312,  4617654272,  4197867520,  2442395648,   888143872,  184549376,   16777216;
  9: 2357947691, 15433839432, 44898441984, 76191295488, 83117776896, 60449292288, 29308747776, 9135194112, 1660944384, 134217728;

%F Formula

The row polynomials satisfy P(n,x) = (11+8*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 + 8*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 + 8*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-b1ed3dc68c32d118
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-6bd8c236b8a4d918, e-b1ed3dc68c32d118
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-6bd8c236b8a4d918, e-b1ed3dc68c32d118
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-b1ed3dc68c32d118
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-6bd8c236b8a4d918
Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.

by the cited steps: e-d06b198556341950
Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 11] to roots in (-inf, 0]: an input root r <= 11 gives the root (r - a)/d <= 0; a larger root gives a positive root
The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.

by the cited steps: e-d06b198556341950

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map_theorem e-d06b198556341950 root: theorem check

Geometric basis map (elementary) polytool ?
parent_provers periodic_product e-6bd8c236b8a4d918 root: theorem check

Parent-row provers of polytool polytool ?
theorem_check liu_wang e-b1ed3dc68c32d118 root: theorem check

Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 7 proof steps

e-36980f01022d8136 basis_map (A013611) → basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-6db2596ebcec1df1 basis_map (A013609) → basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-7c74a26235b2e89e basis_map (A008277) → basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-c22af348c5e552a2 basis_map (A007318) → basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-f9ff39da4c6bf2a9 basis_map (A013610) → basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-991e7bd9709a3127 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
e-991e7bd9709a3127 obreschkoff_combination → obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.

Related sequences ?

Same fingerprint (1)

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

A038226 → basis A007318

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