Real-rooted atlas

A106800 → reverse (shift +1) → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence
\(\left(1 + x\right)\,P_{n}(x) = \left(2\,x + 3\,x^{2} + x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20123
3015511
40110231549
501197610952257
601362245315442031539
7016963122043641287687710299

Show all 15 stored rows

Generated b-files: A004211 (row sums), A059340 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-95b4aa73235b5230 · revision 1 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-629d13292f83019a

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): no

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: 0, 1;
  2: 0, 1,   2;
  3: 0, 1,   5,    5;
  4: 0, 1,  10,   23,    15;
  5: 0, 1,  19,   76,   109,     52;
  6: 0, 1,  36,  224,   531,    544,    203;
  7: 0, 1,  69,  631,  2204,   3641,   2876,    877;
  8: 0, 1, 134, 1749,  8471,  20089,  25208,  16113,  4140;
  9: 0, 1, 263, 4838, 31331, 100171, 177631, 178564, 95495, 21147;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -((n-2)*x+(2*n-3)*x^2+(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx 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[(-((n - 2)*x + (2*n - 3)*x^2 + (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], 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] = (-((n - 2)*x + (2*n - 3)*x^2 + (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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.

A059340: reversed triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1:      1;
   2:      2,      1;
   3:      5,      5,       1;
   4:     15,     23,      10,       1;
   5:     52,    109,      76,      19,       1;
   6:    203,    544,     531,     224,      36,      1;
   7:    877,   2876,    3641,    2204,     631,     69,      1;
   8:   4140,  16113,   25208,   20089,    8471,   1749,    134,     1;
   9:  21147,  95495,  178564,  177631,  100171,  31331,   4838,   263,   1;
  10: 115975, 597155, 1302451, 1559788, 1109473, 471827, 113807, 13454, 520, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (1+x)*P(n,x) = -((n-2)*x+(2*n-3)*x^2+(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 2)*x + (2*n - 3)*x^2 + (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (-((n - 2)*x + (2*n - 3)*x^2 + (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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-ef1c37b6ab776d5a
  • 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-ef1c37b6ab776d5a
  • 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-ef1c37b6ab776d5a

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 (5)

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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741