Real-rooted atlas

A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.

A038335 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 A038335
\(P_{n}(x) = \left(12 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112921
214421681441
31728388829167299261
4207366220869984349926561194481
524883293312013996801049760393660590494084101
6298598413436928251942402519424014171760425152853144185766121
7358318081881169924232632325290790403968092801785641764464104447829691801088541
842998169625798901766772211712101583175689523422720571405363221427701124591650244304672137822859361
951597803523482851737610448555212818284971622420570593075215427944806477139724032247949112964649045868387420489794280046581
10619173642244643802316801567283281920313456656384041141186150403702706753536231419172096099179645184027894275208046490458680348678440116679880978201
11743008370688612981905817622986821468160517203483033607758052245504081459548577792610946614333443272928291072012273481091520306837027288046025554093231381059609350277500542221
128916100448256802449040343043310102291415048275255728537601396449404190720167573928502886414662718744002569426033478287364418453192947201472817730982403313839894710445188725836962824295364817355827511386641
131069932053790721043183752445952469432688600678412909398936518656242051230059724803267691605806284832676916058062848245076870435471361378557396199526457439891508313601723196745249408352472061528288440590076910362541865828329154472377739119461
141283918464548864134811438777630726572057640409497619716172921228492840664606650033766460996909975050649668621523721931980858818448904513126438599607093586739219299803546793369672374263300475136197384354455841283700956646047024427033459159272228767924549613243919932521508681

Generated b-files: A009965 (row sums), A038302 (reversed triangle), A038335 ?

Computed in your browser ?

Technical details
Node id
n-f485cd65a8f8089f · revision 1 · JSON
Definition source
oeis_catalog:A038335
Fingerprint
f6-c91d5d506c107735

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:         12,           9;
  2:        144,         216,           81;
  3:       1728,        3888,         2916,          729;
  4:      20736,       62208,        69984,        34992,         6561;
  5:     248832,      933120,      1399680,      1049760,       393660,        59049;
  6:    2985984,    13436928,     25194240,     25194240,     14171760,      4251528,      531441;
  7:   35831808,   188116992,    423263232,    529079040,    396809280,    178564176,    44641044,     4782969;
  8:  429981696,  2579890176,   6772211712,  10158317568,   9523422720,   5714053632,  2142770112,   459165024,   43046721;
  9: 5159780352, 34828517376, 104485552128, 182849716224, 205705930752, 154279448064, 77139724032, 24794911296, 4649045868, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (12+9*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[(12 + 9*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] = (12 + 9*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.

A038302: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         9,         12;
  2:        81,        216,         144;
  3:       729,       2916,        3888,        1728;
  4:      6561,      34992,       69984,       62208,        20736;
  5:     59049,     393660,     1049760,     1399680,       933120,       248832;
  6:    531441,    4251528,    14171760,    25194240,     25194240,     13436928,      2985984;
  7:   4782969,   44641044,   178564176,   396809280,    529079040,    423263232,    188116992,     35831808;
  8:  43046721,  459165024,  2142770112,  5714053632,   9523422720,  10158317568,   6772211712,   2579890176,   429981696;
  9: 387420489, 4649045868, 24794911296, 77139724032, 154279448064, 205705930752, 182849716224, 104485552128, 34828517376, 5159780352;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (12+9*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[(12 + 9*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] = (12 + 9*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.

A038335: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         12,           9;
  2:        144,         216,           81;
  3:       1728,        3888,         2916,          729;
  4:      20736,       62208,        69984,        34992,         6561;
  5:     248832,      933120,      1399680,      1049760,       393660,        59049;
  6:    2985984,    13436928,     25194240,     25194240,     14171760,      4251528,      531441;
  7:   35831808,   188116992,    423263232,    529079040,    396809280,    178564176,    44641044,     4782969;
  8:  429981696,  2579890176,   6772211712,  10158317568,   9523422720,   5714053632,  2142770112,   459165024,   43046721;
  9: 5159780352, 34828517376, 104485552128, 182849716224, 205705930752, 154279448064, 77139724032, 24794911296, 4649045868, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (12+9*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[(12 + 9*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] = (12 + 9*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-03786f96284c3301
  • 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-03786f96284c3301, e-2dcf274e8e5380c5
  • 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-03786f96284c3301, e-2dcf274e8e5380c5
  • 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-03786f96284c3301
  • 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-2dcf274e8e5380c5
  • 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-2d94fae2f976232e
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 12] to roots in (-inf, 0]: an input root r <= 12 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-2d94fae2f976232e

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