Real-rooted atlas

A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.

A038279 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 A038279
\(P_{n}(x) = \left(8 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1819
26416181
3512192241729
4409620483843216561
53276820480512064040159049
62621441966086144010240960481531441
7209715218350086881281433601792013445614782969
816777216167772167340032183500828672028672179264143046721
913421772815099494475497472220200964128768516096430082304721387420489
10107374182413421772807549747202516582405505024082575368601606144028808013486784401
118589934592118111600647381975040276824064069206016012111052815138816135168084480352088131381059609
12687194767361030792151047086696038429527900160830472192016609443842422210562595225620275201126404224961282429536481
1354975581388889335319756867001489817630709016166495965675520215922769923598712832449839104421724162928640146432499210412541865828329
14439804651110476965813944326253472382976312673619148810748155658242687038914565038197964871974256647872184326560153641000961863685824112122876792454961

Generated b-files: A001019 (row sums), A013615 (reversed triangle), A038279, A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-366b2525683554ec · revision 2 · JSON
Definition source
oeis_catalog:A038279
Fingerprint
f6-8ca1e631330e3f91

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:         8,         1;
  2:        64,        16,        1;
  3:       512,       192,       24,        1;
  4:      4096,      2048,      384,       32,       1;
  5:     32768,     20480,     5120,      640,      40,      1;
  6:    262144,    196608,    61440,    10240,     960,     48,     1;
  7:   2097152,   1835008,   688128,   143360,   17920,   1344,    56,    1;
  8:  16777216,  16777216,  7340032,  1835008,  286720,  28672,  1792,   64,  1;
  9: 134217728, 150994944, 75497472, 22020096, 4128768, 516096, 43008, 2304, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8+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[(8 + 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] = (8 + 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.

A013615: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  8;
  2: 1, 16,   64;
  3: 1, 24,  192,   512;
  4: 1, 32,  384,  2048,   4096;
  5: 1, 40,  640,  5120,  20480,   32768;
  6: 1, 48,  960, 10240,  61440,  196608,   262144;
  7: 1, 56, 1344, 17920, 143360,  688128,  1835008,  2097152;
  8: 1, 64, 1792, 28672, 286720, 1835008,  7340032, 16777216,  16777216;
  9: 1, 72, 2304, 43008, 516096, 4128768, 22020096, 75497472, 150994944, 134217728;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (8+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[(8 + 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] = (8 + 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.

A038279: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:         1;
  1:         8,         1;
  2:        64,        16,        1;
  3:       512,       192,       24,        1;
  4:      4096,      2048,      384,       32,       1;
  5:     32768,     20480,     5120,      640,      40,      1;
  6:    262144,    196608,    61440,    10240,     960,     48,     1;
  7:   2097152,   1835008,   688128,   143360,   17920,   1344,    56,    1;
  8:  16777216,  16777216,  7340032,  1835008,  286720,  28672,  1792,   64,  1;
  9: 134217728, 150994944, 75497472, 22020096, 4128768, 516096, 43008, 2304, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8+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[(8 + 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] = (8 + 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-b6a9810385e1026e
  • 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-403b5075da2bf036, e-b6a9810385e1026e, e-e44a3cf5ca807197
  • 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-403b5075da2bf036, e-b6a9810385e1026e, e-e44a3cf5ca807197
  • 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-b6a9810385e1026e
  • 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-403b5075da2bf036, e-e44a3cf5ca807197
  • 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-e806b1e60392ec27
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 8] to roots in (-inf, 0]: an input root r <= 8 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-e806b1e60392ec27

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