Real-rooted atlas

A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.

A038221 palindromic γ-positive 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 A038221
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2918936
327818127216
481324486324811296
524312152430243012152437776
67294374109351458010935437472946656
721871530945927765457654545927153092187279936

Show all 15 stored rows

Generated b-files: A000400 (row sums), A038221, A038221 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-8d88ef9de74b305f · revision 2 · JSON
Definition source
oeis_catalog:A038221
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:     3,      3;
  2:     9,     18,      9;
  3:    27,     81,     81,      27;
  4:    81,    324,    486,     324,      81;
  5:   243,   1215,   2430,    2430,    1215,     243;
  6:   729,   4374,  10935,   14580,   10935,    4374,     729;
  7:  2187,  15309,  45927,   76545,   76545,   45927,   15309,   2187;
  8:  6561,  52488, 183708,  367416,  459270,  367416,  183708,  52488,   6561;
  9: 19683, 177147, 708588, 1653372, 2480058, 2480058, 1653372, 708588, 177147, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*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[(3 + 3*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] = (3 + 3*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.

A038221: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      3;
  2:     9,     18,      9;
  3:    27,     81,     81,      27;
  4:    81,    324,    486,     324,      81;
  5:   243,   1215,   2430,    2430,    1215,     243;
  6:   729,   4374,  10935,   14580,   10935,    4374,     729;
  7:  2187,  15309,  45927,   76545,   76545,   45927,   15309,   2187;
  8:  6561,  52488, 183708,  367416,  459270,  367416,  183708,  52488,   6561;
  9: 19683, 177147, 708588, 1653372, 2480058, 2480058, 1653372, 708588, 177147, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*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[(3 + 3*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] = (3 + 3*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.

A038221: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      3;
  2:     9,     18,      9;
  3:    27,     81,     81,      27;
  4:    81,    324,    486,     324,      81;
  5:   243,   1215,   2430,    2430,    1215,     243;
  6:   729,   4374,  10935,   14580,   10935,    4374,     729;
  7:  2187,  15309,  45927,   76545,   76545,   45927,   15309,   2187;
  8:  6561,  52488, 183708,  367416,  459270,  367416,  183708,  52488,   6561;
  9: 19683, 177147, 708588, 1653372, 2480058, 2480058, 1653372, 708588, 177147, 19683;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*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[(3 + 3*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] = (3 + 3*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-b7eead465111f7d4
  • 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-110e02c529078258, e-b7eead465111f7d4, e-d377a44ea8f9b38f
  • 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-110e02c529078258, e-b7eead465111f7d4, e-d377a44ea8f9b38f
  • 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-b7eead465111f7d4
  • 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-110e02c529078258, e-d377a44ea8f9b38f
  • 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-98d48bf052252a89
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 3] to roots in (-inf, 0]: an input root r <= 3 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-98d48bf052252a89

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 8 proof steps

Same fingerprint (59)

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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [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