Real-rooted atlas

A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).

A027466 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 A027466
\(P_{n}(x) = \left(7 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1718
24914164
3343147211512
4240113722942814096
51680712005343049035132768
6117649100842360156860735421262144
7823543823543352947840351200510294912097152
857648016588344329417294119216807019208137256116777216
940353607518832092964754898825162117682302526288121764631134217728
10282475249403536070259416045988251602470629042353645042104116022057011073741824
111977326743310722773922194483859511921652717691905435383877648347923305659526957718589934592
121384128720123727920916186433664348877793540285357649565224605610870767613311144118849575460323484168719476736
1396889010407179936733613154231485954807879212142885282900574192988871413199788201885684216306091716715980983822911549755813888
14678223072849135644614569812595571352917197469344522827577242498078792121417311697403282639957635329994733647614240340112485244599814398046511104

Generated b-files: A001018 (row sums), A013614 (reversed triangle), A027466 ?

Computed in your browser ?

Technical details
Node id
n-f211cb6153c37c6d · revision 2 · JSON
Definition source
oeis_catalog:A027466
Fingerprint
f6-d2d514e02d66ef29

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:        7,        1;
  2:       49,       14,        1;
  3:      343,      147,       21,       1;
  4:     2401,     1372,      294,      28,       1;
  5:    16807,    12005,     3430,     490,      35,      1;
  6:   117649,   100842,    36015,    6860,     735,     42,     1;
  7:   823543,   823543,   352947,   84035,   12005,   1029,    49,    1;
  8:  5764801,  6588344,  3294172,  941192,  168070,  19208,  1372,   56,  1;
  9: 40353607, 51883209, 29647548, 9882516, 2117682, 302526, 28812, 1764, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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.

A013614: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  7;
  2: 1, 14,   49;
  3: 1, 21,  147,   343;
  4: 1, 28,  294,  1372,   2401;
  5: 1, 35,  490,  3430,  12005,   16807;
  6: 1, 42,  735,  6860,  36015,  100842,  117649;
  7: 1, 49, 1029, 12005,  84035,  352947,  823543,   823543;
  8: 1, 56, 1372, 19208, 168070,  941192, 3294172,  6588344,  5764801;
  9: 1, 63, 1764, 28812, 302526, 2117682, 9882516, 29647548, 51883209, 40353607;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+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[(7 + 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] = (7 + 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.

A027466: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:        1;
  1:        7,        1;
  2:       49,       14,        1;
  3:      343,      147,       21,       1;
  4:     2401,     1372,      294,      28,       1;
  5:    16807,    12005,     3430,     490,      35,      1;
  6:   117649,   100842,    36015,    6860,     735,     42,     1;
  7:   823543,   823543,   352947,   84035,   12005,   1029,    49,    1;
  8:  5764801,  6588344,  3294172,  941192,  168070,  19208,  1372,   56,  1;
  9: 40353607, 51883209, 29647548, 9882516, 2117682, 302526, 28812, 1764, 63, 1;
%F Formula
The row polynomials satisfy P(n,x) = (7+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[(7 + 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] = (7 + 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-c0e43064b314c6c0
  • 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-7e8833b30e1c308e, e-c0e43064b314c6c0, e-cffbbf28e801bdbf
  • 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-7e8833b30e1c308e, e-c0e43064b314c6c0, e-cffbbf28e801bdbf
  • 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-c0e43064b314c6c0
  • 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-7e8833b30e1c308e, e-cffbbf28e801bdbf
  • 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-bbe2611159422330
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 7] to roots in (-inf, 0]: an input root r <= 7 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-bbe2611159422330

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

Same fingerprint (1)

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