Real-rooted atlas

A008957 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 4\,x + 7\,x^{2} + 9\,x^{3} + 10\,x^{4} + 7\,x^{5} + 2\,x^{6} - 3\,n x^{2} - 12\,n x^{3} - 18\,n x^{4} - 12\,n x^{5} - 3\,n x^{6} + n^{2} x + 5\,n^{2} x^{2} + 10\,n^{2} x^{3} + 10\,n^{2} x^{4} + 5\,n^{2} x^{5} + n^{2} x^{6}\right)\,P_{n-1}(x) + \left(x^{2} + 8\,x^{3} + 22\,x^{4} + 28\,x^{5} + 17\,x^{6} + 4\,x^{7} - 2\,n x^{2} - 10\,n x^{3} - 20\,n x^{4} - 20\,n x^{5} - 10\,n x^{6} - 2\,n x^{7}\right)\,P_{n-1}'(x) + \left(x^{3} + 5\,x^{4} + 10\,x^{5} + 10\,x^{6} + 5\,x^{7} + x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
217715
31175237107
41342434732641015
516085736295263243312243
6197247220378602707018027913181311
71147613990015497628113048211037813869063215099

All 15 rows

Technical details
Node id
n-8bbb36eb39e7ea68 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-73fe859bd5ab53a7

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,   2;
  2: 1,   7,     7;
  3: 1,  17,    52,      37;
  4: 1,  34,   243,     473,      264;
  5: 1,  60,   857,    3629,     5263,      2433;
  6: 1,  97,  2472,   20378,    60270,     70180,     27913;
  7: 1, 147,  6139,   90015,   497628,   1130482,   1103781,    386906;
  8: 1, 212, 13594,  328666,  3180073,  12953478,  23938969,  20196399,   6346119;
  9: 1, 294, 27510, 1030682, 16409951, 114119826, 365204043, 569770003, 424438672, 121159373;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+(n^2+4)*x+(5*n^2-3*n+7)*x^2+(10*n^2-12*n+9)*x^3+(10*n^2-18*n+10)*x^4+(5*n^2-12*n+7)*x^5+(n^2-3*n+2)*x^6)*P(n-1,x) - ((2*n-1)*x^2+(10*n-8)*x^3+(20*n-22)*x^4+(20*n-28)*x^5+(10*n-17)*x^6+(2*n-4)*x^7)*d/dx P(n-1,x) + (x^3+5*x^4+10*x^5+10*x^6+5*x^7+x^8)*d^2/dx^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[((1 + (n^2 + 4)*x + (5*n^2 - 3*n + 7)*x^2 + (10*n^2 - 12*n + 9)*x^3 + (10*n^2 - 18*n + 10)*x^4 + (5*n^2 - 12*n + 7)*x^5 + (n^2 - 3*n + 2)*x^6)*P[n - 1, x] - ((2*n - 1)*x^2 + (10*n - 8)*x^3 + (20*n - 22)*x^4 + (20*n - 28)*x^5 + (10*n - 17)*x^6 + (2*n - 4)*x^7)*D[P[n - 1, x], x] + (x^3 + 5*x^4 + 10*x^5 + 10*x^6 + 5*x^7 + x^8)*D[P[n - 1, x], {x, 2}])/(1 + 3*x + 3*x^2 + x^3)]];
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] = ((1 + (n^2 + 4)*x + (5*n^2 - 3*n + 7)*x^2 + (10*n^2 - 12*n + 9)*x^3 + (10*n^2 - 18*n + 10)*x^4 + (5*n^2 - 12*n + 7)*x^5 + (n^2 - 3*n + 2)*x^6)*v[i - 1] - ((2*n - 1)*x^2 + (10*n - 8)*x^3 + (20*n - 22)*x^4 + (20*n - 28)*x^5 + (10*n - 17)*x^6 + (2*n - 4)*x^7)*deriv(v[i - 1], x) + (x^3 + 5*x^4 + 10*x^5 + 10*x^6 + 5*x^7 + x^8)*deriv(deriv(v[i - 1], x), x))/(1 + 3*x + 3*x^2 + x^3)); 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

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.