Real-rooted atlas

A038208 → x ↦ x/(x+1)

nonnegative coefficients only rational roots

Proved for all n: all roots real, in [−1, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 6\,x + 4\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
24161636
38489664216
4161283845122561296
53232012802560256010247776
6647683840102401536012288409646656
71281792107523584071680860165734416384279936
8256409628672114688286720458752458752262144655361679616
95129216737283440641032192206438427525122359296117964826214410077696
10102420480184320983040344064082575361376256015728640117964805242880104857660466176
112048450564505602703360108134403027763260555264865075208650752057671680230686724194304362797056
124096983041081344720896032440320103809024242221056415236096519045120461373440276824064100663296167772162176782336
13819221299225559041874329693716480337379328899678208179935641626990346242998927360239914188813086228484362076166710886413060694016
1416384458752596377647710208262406144104962457631488737287197425664125954949121679399321616793993216122138132486106906624187904819226843545678364164096

Generated b-files: A000400 (row sums), A038210, A038232 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-8f1f80f146f45463 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-11aec07ce60b6d61

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:   2,    4;
  2:   4,   16,    16;
  3:   8,   48,    96,     64;
  4:  16,  128,   384,    512,     256;
  5:  32,  320,  1280,   2560,    2560,    1024;
  6:  64,  768,  3840,  10240,   15360,   12288,    4096;
  7: 128, 1792, 10752,  35840,   71680,   86016,   57344,   16384;
  8: 256, 4096, 28672, 114688,  286720,  458752,  458752,  262144,   65536;
  9: 512, 9216, 73728, 344064, 1032192, 2064384, 2752512, 2359296, 1179648, 262144;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2+6*x+4*x^2)*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[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(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] = ((2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); 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.

A038210: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    4;
  2:   4,   16,    16;
  3:   8,   48,    96,     64;
  4:  16,  128,   384,    512,     256;
  5:  32,  320,  1280,   2560,    2560,    1024;
  6:  64,  768,  3840,  10240,   15360,   12288,    4096;
  7: 128, 1792, 10752,  35840,   71680,   86016,   57344,   16384;
  8: 256, 4096, 28672, 114688,  286720,  458752,  458752,  262144,   65536;
  9: 512, 9216, 73728, 344064, 1032192, 2064384, 2752512, 2359296, 1179648, 262144;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2+6*x+4*x^2)*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[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(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] = ((2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); 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.

A038232: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       2;
  2:     16,      16,       4;
  3:     64,      96,      48,       8;
  4:    256,     512,     384,     128,      16;
  5:   1024,    2560,    2560,    1280,     320,      32;
  6:   4096,   12288,   15360,   10240,    3840,     768,     64;
  7:  16384,   57344,   86016,   71680,   35840,   10752,   1792,   128;
  8:  65536,  262144,  458752,  458752,  286720,  114688,  28672,  4096,  256;
  9: 262144, 1179648, 2359296, 2752512, 2064384, 1032192, 344064, 73728, 9216, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (1+x)*P(n,x) = (2+6*x+4*x^2)*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[Together[((2 + 6*x + 4*x^2)*P[n - 1, x])/(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] = ((2 + 6*x + 4*x^2)*v[i - 1])/(1 + x)); 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 [-1, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-19854cf600900932
  • 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-19854cf600900932
  • 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-19854cf600900932

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

Same fingerprint (66)

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