Real-rooted atlas

A013620 → 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
\(\left(1 + x\right)\,P_{n}(x) = \left(2 + 7\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
24202549
3860150125343
41616060010006252401
532400200050006250312516807
664960600020000375003750015625117649
71282240168007000017500026250021875078125823543

Show all 15 stored rows

Generated b-files: A000420 (row sums), A013621, A038244 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-f1a275eb676f791b · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9629d1e9c04dbfbe

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,     5;
  2:   4,    20,     25;
  3:   8,    60,    150,    125;
  4:  16,   160,    600,   1000,     625;
  5:  32,   400,   2000,   5000,    6250,    3125;
  6:  64,   960,   6000,  20000,   37500,   37500,    15625;
  7: 128,  2240,  16800,  70000,  175000,  262500,   218750,    78125;
  8: 256,  5120,  44800, 224000,  700000, 1400000,  1750000,  1250000,  390625;
  9: 512, 11520, 115200, 672000, 2520000, 6300000, 10500000, 11250000, 7031250, 1953125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2+7*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[((2 + 7*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] = ((2 + 7*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.

A013621: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,     5;
  2:   4,    20,     25;
  3:   8,    60,    150,    125;
  4:  16,   160,    600,   1000,     625;
  5:  32,   400,   2000,   5000,    6250,    3125;
  6:  64,   960,   6000,  20000,   37500,   37500,    15625;
  7: 128,  2240,  16800,  70000,  175000,  262500,   218750,    78125;
  8: 256,  5120,  44800, 224000,  700000, 1400000,  1750000,  1250000,  390625;
  9: 512, 11520, 115200, 672000, 2520000, 6300000, 10500000, 11250000, 7031250, 1953125;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (2+7*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[((2 + 7*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] = ((2 + 7*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.

A038244: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:       1;
  1:       5,       2;
  2:      25,      20,        4;
  3:     125,     150,       60,        8;
  4:     625,    1000,      600,      160,      16;
  5:    3125,    6250,     5000,     2000,     400,      32;
  6:   15625,   37500,    37500,    20000,    6000,     960,     64;
  7:   78125,  218750,   262500,   175000,   70000,   16800,   2240,    128;
  8:  390625, 1250000,  1750000,  1400000,  700000,  224000,  44800,   5120,   256;
  9: 1953125, 7031250, 11250000, 10500000, 6300000, 2520000, 672000, 115200, 11520, 512;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (1+x)*P(n,x) = (2+7*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[((2 + 7*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] = ((2 + 7*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-867523535b04b58e
  • 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-867523535b04b58e
  • 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-867523535b04b58e

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