Real-rooted atlas

A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.

A038259 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 A038259
\(P_{n}(x) = \left(6 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16511
2366025121
32165404501251331
4129643205400300062514641
57776324005400045000187503125161051
646656233280486000540000337500112500156251771561
7279936163296040824005670000472500023625006562507812519487171

Show all 15 stored rows

Generated b-files: A001020 (row sums), A038248 (reversed triangle), A038259 ?

Computed in your browser ?

Technical details
Node id
n-e6ebe74518d42cbe · revision 1 · JSON
Definition source
oeis_catalog:A038259
Fingerprint
f6-da2debf48cb6165b

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:        6,        5;
  2:       36,       60,        25;
  3:      216,      540,       450,       125;
  4:     1296,     4320,      5400,      3000,       625;
  5:     7776,    32400,     54000,     45000,     18750,      3125;
  6:    46656,   233280,    486000,    540000,    337500,    112500,     15625;
  7:   279936,  1632960,   4082400,   5670000,   4725000,   2362500,    656250,     78125;
  8:  1679616, 11197440,  32659200,  54432000,  56700000,  37800000,  15750000,   3750000,   390625;
  9: 10077696, 75582720, 251942400, 489888000, 612360000, 510300000, 283500000, 101250000, 21093750, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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.

A038248: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        6;
  2:      25,       60,        36;
  3:     125,      450,       540,       216;
  4:     625,     3000,      5400,      4320,      1296;
  5:    3125,    18750,     45000,     54000,     32400,      7776;
  6:   15625,   112500,    337500,    540000,    486000,    233280,     46656;
  7:   78125,   656250,   2362500,   4725000,   5670000,   4082400,   1632960,    279936;
  8:  390625,  3750000,  15750000,  37800000,  56700000,  54432000,  32659200,  11197440,  1679616;
  9: 1953125, 21093750, 101250000, 283500000, 510300000, 612360000, 489888000, 251942400, 75582720, 10077696;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+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[(6 + 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] = (6 + 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.

A038259: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        6,        5;
  2:       36,       60,        25;
  3:      216,      540,       450,       125;
  4:     1296,     4320,      5400,      3000,       625;
  5:     7776,    32400,     54000,     45000,     18750,      3125;
  6:    46656,   233280,    486000,    540000,    337500,    112500,     15625;
  7:   279936,  1632960,   4082400,   5670000,   4725000,   2362500,    656250,     78125;
  8:  1679616, 11197440,  32659200,  54432000,  56700000,  37800000,  15750000,   3750000,   390625;
  9: 10077696, 75582720, 251942400, 489888000, 612360000, 510300000, 283500000, 101250000, 21093750, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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-0a7984b7b9125974
  • 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-0a7984b7b9125974, e-7949976749bfedcf
  • 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-0a7984b7b9125974, e-7949976749bfedcf
  • 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-0a7984b7b9125974
  • 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-7949976749bfedcf
  • 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-34a032c211197635
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 6] to roots in (-inf, 0]: an input root r <= 6 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-34a032c211197635

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 (6)

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