Real-rooted atlas

A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.

A038277 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 A038277
\(P_{n}(x) = \left(7 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
171118
249154121324
33431617254113315832
4240115092355743726814641104976
5168071320554150306521905124351610511889568
6117649110926243578159130660107611356764142177156134012224
78235439058973427065871118505851757652051657214798680648919487171612220032

Show all 15 stored rows

Generated b-files: A001027 (row sums), A038277, A038321 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-b7cf68162d40a956 · revision 1 · JSON
Definition source
oeis_catalog:A038277
Fingerprint
f6-428cb055b0457fc7

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 ?

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:        7,        11;
  2:       49,       154,        121;
  3:      343,      1617,       2541,        1331;
  4:     2401,     15092,      35574,       37268,       14641;
  5:    16807,    132055,     415030,      652190,      512435,      161051;
  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;
  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;
  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;
  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (7+11*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[(7 + 11*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] = (7 + 11*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.

A038277: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,        11;
  2:       49,       154,        121;
  3:      343,      1617,       2541,        1331;
  4:     2401,     15092,      35574,       37268,       14641;
  5:    16807,    132055,     415030,      652190,      512435,      161051;
  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;
  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;
  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;
  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (7+11*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[(7 + 11*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] = (7 + 11*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.

A038321: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,           7;
  2:        121,         154,          49;
  3:       1331,        2541,        1617,         343;
  4:      14641,       37268,       35574,       15092,        2401;
  5:     161051,      512435,      652190,      415030,      132055,       16807;
  6:    1771561,     6764142,    10761135,     9130660,     4357815,     1109262,      117649;
  7:   19487171,    86806489,   165721479,   175765205,   111850585,    42706587,     9058973,     823543;
  8:  214358881,  1091281576,  2430581692,  3093467608,  2460712870,  1252726552,   398594812,   72471784,   5764801;
  9: 2357947691, 13504609503, 34375369644, 51042215532, 48722114826, 31004982162, 13153628796, 3587353308, 570715299, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+11*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[(7 + 11*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] = (7 + 11*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.

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-f9e3d73ad0eb8fa6
  • 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-399c4cc80f25ee4f, e-f9e3d73ad0eb8fa6
  • 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-399c4cc80f25ee4f, e-f9e3d73ad0eb8fa6
  • 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-f9e3d73ad0eb8fa6
  • 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-399c4cc80f25ee4f
  • 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-dc6cd425a1850984
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 7] to roots in (-inf, 0]: an input root r <= 7 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-dc6cd425a1850984

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

Used in 7 proof steps

Same fingerprint (1)

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

References

  1. [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