Real-rooted atlas

A013622: Triangle of coefficients in expansion of (3+5x)^n.

A013622 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · all roots real, in [−1, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A013622
\(P_{n}(x) = \left(3 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
29302564
327135225125512
481540135015006254096
524320256750112509375312532768
672972903037567500843755625015625262144
7218725515127575354375590625590625328125781252097152

Show all 15 stored rows

Generated b-files: A001018 (row sums), A013622, A038245 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x/(x+1) substitution · proved equal for all n (proof step)

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 8\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-c44f7114310405ad · revision 2 · JSON
Definition source
oeis_catalog:A013622
derived:mobius_x_over_x_plus_one (equal by e-0961180c91a1e61b)
Fingerprint
f6-b1029f17d1b64336

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:     3,      5;
  2:     9,     30,      25;
  3:    27,    135,     225,     125;
  4:    81,    540,    1350,    1500,      625;
  5:   243,   2025,    6750,   11250,     9375,     3125;
  6:   729,   7290,   30375,   67500,    84375,    56250,    15625;
  7:  2187,  25515,  127575,  354375,   590625,   590625,   328125,    78125;
  8:  6561,  87480,  510300, 1701000,  3543750,  4725000,  3937500,  1875000,   390625;
  9: 19683, 295245, 1968300, 7654500, 19136250, 31893750, 35437500, 25312500, 10546875, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (3+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[(3 + 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] = (3 + 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.

A013622: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:     1;
  1:     3,      5;
  2:     9,     30,      25;
  3:    27,    135,     225,     125;
  4:    81,    540,    1350,    1500,      625;
  5:   243,   2025,    6750,   11250,     9375,     3125;
  6:   729,   7290,   30375,   67500,    84375,    56250,    15625;
  7:  2187,  25515,  127575,  354375,   590625,   590625,   328125,    78125;
  8:  6561,  87480,  510300, 1701000,  3543750,  4725000,  3937500,  1875000,   390625;
  9: 19683, 295245, 1968300, 7654500, 19136250, 31893750, 35437500, 25312500, 10546875, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (3+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[(3 + 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] = (3 + 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.

A038245: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        3;
  2:      25,       30,        9;
  3:     125,      225,      135,       27;
  4:     625,     1500,     1350,      540,       81;
  5:    3125,     9375,    11250,     6750,     2025,      243;
  6:   15625,    56250,    84375,    67500,    30375,     7290,     729;
  7:   78125,   328125,   590625,   590625,   354375,   127575,   25515,    2187;
  8:  390625,  1875000,  3937500,  4725000,  3543750,  1701000,  510300,   87480,   6561;
  9: 1953125, 10546875, 25312500, 35437500, 31893750, 19136250, 7654500, 1968300, 295245, 19683;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+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[(3 + 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] = (3 + 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.

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-9aec8f9af8bb24ea
  • 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-9aec8f9af8bb24ea, e-a28ac028c491cc22, e-abc3ddb24ba116e8
  • 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-9aec8f9af8bb24ea, e-a28ac028c491cc22, e-abc3ddb24ba116e8
  • 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-9aec8f9af8bb24ea
  • 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-abc3ddb24ba116e8
  • 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-60a54feceb89d236
  • 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-60a54feceb89d236
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a28ac028c491cc22

Proof chains

5 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 9 proof steps

Same fingerprint (4)

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