Real-rooted atlas

A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).

A027467 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 A027467
\(P_{n}(x) = \left(15 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115116
2225301256
333756754514096
45062513500135060165536
57593752531253375022507511048576
611390625455625075937567500337590116777216
71708593757973437515946875177187511812547251051268435456
825628906251366875000318937500425250003543750189000630012014294967296
9384433593752306601562561509375009568125009568125063787502835008100135168719476736
10576650390625384433593750115330078125205031250002392031250191362500106312504050001012515011099511627776
118649755859375634315429687521143847656254228769531255638359375052624687503508312501670625055687512375165117592186044416
1212974633789062510379707031250038058925781250845753906250012686308593751353206250001052493750060142500025059375742500148501801281474976710656
131946195068359375168670239257812567468095703125016492201171875027487001953125329844023437529319468750019546312500977315625361968759652501755019514503599627370496
14291929260253906252724673095703125011806916748046875314851113281250057722704101562576963605468750769636054687558638937500034206046875152026875050675625122850020475210172057594037927936

Generated b-files: A001025 (row sums), A027467 ?

Computed in your browser ?

Technical details
Node id
n-ff0214c333bfdcca · revision 2 · JSON
Definition source
oeis_catalog:A027467
Fingerprint
f6-8deca67d09eebe9e

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:          15,           1;
  2:         225,          30,          1;
  3:        3375,         675,         45,         1;
  4:       50625,       13500,       1350,        60,        1;
  5:      759375,      253125,      33750,      2250,       75,       1;
  6:    11390625,     4556250,     759375,     67500,     3375,      90,      1;
  7:   170859375,    79734375,   15946875,   1771875,   118125,    4725,    105,    1;
  8:  2562890625,  1366875000,  318937500,  42525000,  3543750,  189000,   6300,  120,   1;
  9: 38443359375, 23066015625, 6150937500, 956812500, 95681250, 6378750, 283500, 8100, 135, 1;
%F Formula
The row polynomials satisfy P(n,x) = (15+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[(15 + 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] = (15 + 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.

A027467: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:           1;
  1:          15,           1;
  2:         225,          30,          1;
  3:        3375,         675,         45,         1;
  4:       50625,       13500,       1350,        60,        1;
  5:      759375,      253125,      33750,      2250,       75,       1;
  6:    11390625,     4556250,     759375,     67500,     3375,      90,      1;
  7:   170859375,    79734375,   15946875,   1771875,   118125,    4725,    105,    1;
  8:  2562890625,  1366875000,  318937500,  42525000,  3543750,  189000,   6300,  120,   1;
  9: 38443359375, 23066015625, 6150937500, 956812500, 95681250, 6378750, 283500, 8100, 135, 1;
%F Formula
The row polynomials satisfy P(n,x) = (15+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[(15 + 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] = (15 + 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-14b8bc50df3e26b6
  • 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-14b8bc50df3e26b6, e-5234407c4fbd8679, e-7eb5590197b773bf
  • 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-14b8bc50df3e26b6, e-5234407c4fbd8679, e-7eb5590197b773bf
  • 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-14b8bc50df3e26b6
  • 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-5234407c4fbd8679, e-7eb5590197b773bf
  • 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-d5b8d319fcb3f7e8
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 15] to roots in (-inf, 0]: an input root r <= 15 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-d5b8d319fcb3f7e8

Proof chains

4 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

No candidate equalities or related sequences.

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