Real-rooted atlas

A166345: Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.

A166345 ↗ palindromicγ-positivenonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)interlacing

Recurrence from A166345

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21214
3177116
41184218180
5141198198411480
618879915847998813360
71183292510331103312925183126880

All 15 rows b-files: A166345, A166345 (reversed triangle) ?

Technical details
Node id
n-5c39e26cb9304567 · revision 0 · JSON
Definition source
oeis_catalog:A166345
Fingerprint
f6-c9655e7f89d7b43e

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

  • A166345 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
  • A166345 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   1;
  2: 1,   2,     1;
  3: 1,   7,     7,      1;
  4: 1,  18,    42,     18,      1;
  5: 1,  41,   198,    198,     41,      1;
  6: 1,  88,   799,   1584,    799,     88,      1;
  7: 1, 183,  2925,  10331,  10331,   2925,    183,     1;
  8: 1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;
  9: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n^2-2*n)*x)*P(n-1,x) + ((n-2)*x-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 2*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n^2 - 2*n)*x)*P[n - 1, x] + ((n - 2)*x - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n^2 - 2*n)*x)*v[i - 1] + ((n - 2)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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)

polytool c48629374597

A166345 triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   2,     1;
   4: 1,   7,     7,      1;
   5: 1,  18,    42,     18,      1;
   6: 1,  41,   198,    198,     41,      1;
   7: 1,  88,   799,   1584,    799,     88,      1;
   8: 1, 183,  2925,  10331,  10331,   2925,    183,     1;
   9: 1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;
  10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A166345 reversed triangle, offset 1, from P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   2,     1;
   4: 1,   7,     7,      1;
   5: 1,  18,    42,     18,      1;
   6: 1,  41,   198,    198,     41,      1;
   7: 1,  88,   799,   1584,    799,     88,      1;
   8: 1, 183,  2925,  10331,  10331,   2925,    183,     1;
   9: 1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;
  10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 2*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); 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)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [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