Real-rooted atlas

A038231 → 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
\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 9\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
216402581
364240300125729
42561280240020006256561
510246400160002000012500312559049
6409630720960001600001500007500015625531441
716384143360537600112000014000001050000437500781254782969
8655366553602867200716800011200000112000007000000250000039062543046721
926214429491201474560043008000806400001008000008400000045000000140625001953125387420489
10104857613107200737280002457600005376000008064000008400000006000000002812500007812500097656253486784401
1141943045767168036044800013516800003379200000591360000073920000006600000000412500000017187500004296875004882812531381059609
121677721625165824017301504007208960000202752000004055040000059136000000633600000004950000000027500000000103125000002343750000244140625282429536481
13671088641090519040817889280037486592000117145600000263577600000439296000000549120000000514800000000357500000000178750000000609375000001269531250012207031252541865828329
14268435456469762048038168166400190840832000656015360000164003840000030750720000004392960000000480480000000040040000000002502500000000113750000000035546875000068359375000610351562522876792454961

Generated b-files: A001019 (row sums), A013628, A038246 (reversed triangle), A100062 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-506262231820dae3 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6b726760bd05165a

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:      4,       5;
  2:     16,      40,       25;
  3:     64,     240,      300,      125;
  4:    256,    1280,     2400,     2000,      625;
  5:   1024,    6400,    16000,    20000,    12500,      3125;
  6:   4096,   30720,    96000,   160000,   150000,     75000,    15625;
  7:  16384,  143360,   537600,  1120000,  1400000,   1050000,   437500,    78125;
  8:  65536,  655360,  2867200,  7168000, 11200000,  11200000,  7000000,  2500000,   390625;
  9: 262144, 2949120, 14745600, 43008000, 80640000, 100800000, 84000000, 45000000, 14062500, 1953125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4+9*x+5*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[((4 + 9*x + 5*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] = ((4 + 9*x + 5*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.

A013628: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      4,       5;
  2:     16,      40,       25;
  3:     64,     240,      300,      125;
  4:    256,    1280,     2400,     2000,      625;
  5:   1024,    6400,    16000,    20000,    12500,      3125;
  6:   4096,   30720,    96000,   160000,   150000,     75000,    15625;
  7:  16384,  143360,   537600,  1120000,  1400000,   1050000,   437500,    78125;
  8:  65536,  655360,  2867200,  7168000, 11200000,  11200000,  7000000,  2500000,   390625;
  9: 262144, 2949120, 14745600, 43008000, 80640000, 100800000, 84000000, 45000000, 14062500, 1953125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4+9*x+5*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[((4 + 9*x + 5*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] = ((4 + 9*x + 5*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.

A038246: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,        4;
  2:      25,       40,       16;
  3:     125,      300,      240,       64;
  4:     625,     2000,     2400,     1280,       256;
  5:    3125,    12500,    20000,    16000,      6400,     1024;
  6:   15625,    75000,   150000,   160000,     96000,    30720,     4096;
  7:   78125,   437500,  1050000,  1400000,   1120000,   537600,   143360,    16384;
  8:  390625,  2500000,  7000000, 11200000,  11200000,  7168000,  2867200,   655360,   65536;
  9: 1953125, 14062500, 45000000, 84000000, 100800000, 80640000, 43008000, 14745600, 2949120, 262144;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (1+x)*P(n,x) = (4+9*x+5*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[((4 + 9*x + 5*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] = ((4 + 9*x + 5*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-7e32de33a5202f3a
  • 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-7e32de33a5202f3a
  • 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-7e32de33a5202f3a

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

Same fingerprint (6)

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