Real-rooted atlas

A028421 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(-1 - x + 7\,x^{2} + 17\,x^{3} + 14\,x^{4} + 4\,x^{5} + 2\,n + 6\,n x + 16\,n x^{2} + 40\,n x^{3} + 54\,n x^{4} + 34\,n x^{5} + 8\,n x^{6} - n^{2} - 5\,n^{2} x - 14\,n^{2} x^{2} - 26\,n^{2} x^{3} - 29\,n^{2} x^{4} - 17\,n^{2} x^{5} - 4\,n^{2} x^{6}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 16\,x^{3} - 24\,x^{4} - 16\,x^{5} - 4\,x^{6} + 4\,n x^{2} + 20\,n x^{3} + 40\,n x^{4} + 40\,n x^{5} + 20\,n x^{6} + 4\,n x^{7}\right)\,P_{n-2}'(x) + \left(-1 + 2\,x^{2} - 4\,x^{3} - 9\,x^{4} - 4\,x^{5} + 2\,n + 8\,n x + 16\,n x^{2} + 20\,n x^{3} + 14\,n x^{4} + 4\,n x^{5}\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 16\,x^{3} - 24\,x^{4} - 16\,x^{5} - 4\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1156
22162341
3662160120348
424296105914867243589
5120169676091521114154501243802
6720113766069615518420871114285039332618869
750408755253766016597242885936288021615446243458329946584

All 15 rows

Technical details
Node id
n-d0dded0d1e8e9e68 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-841a32018555c02e

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,       5;
  2:      2,      16,       23;
  3:      6,      62,      160,       120;
  4:     24,     296,     1059,      1486,       724;
  5:    120,    1696,     7609,     15211,     14154,      5012;
  6:    720,   11376,    60696,    155184,    208711,    142850,     39332;
  7:   5040,   87552,   537660,   1659724,   2885936,   2880216,   1544624,    345832;
  8:  40320,  760896,  5260428,  18918216,  40054845,  52035636,  40930188,  17942928,   3371976;
  9: 362880, 7372224, 56468412, 230917412, 573522609, 912191313, 938318656, 605806740, 223750824, 36135792;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -(n^2-2*n+1+(5*n^2-6*n+1)*x+(14*n^2-16*n-7)*x^2+(26*n^2-40*n-17)*x^3+(29*n^2-54*n-14)*x^4+(17*n^2-34*n-4)*x^5+(4*n^2-8*n)*x^6)*P(n-2,x) + ((4*n-4)*x^2+(20*n-16)*x^3+(40*n-24)*x^4+(40*n-16)*x^5+(20*n-4)*x^6+4*n*x^7)*d/dx P(n-2,x) + (2*n-1+8*n*x+(16*n+2)*x^2+(20*n-4)*x^3+(14*n-9)*x^4+(4*n-4)*x^5)*P(n-1,x) - (4*x^2+16*x^3+24*x^4+16*x^5+4*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 + (5*n^2 - 6*n + 1)*x + (14*n^2 - 16*n - 7)*x^2 + (26*n^2 - 40*n - 17)*x^3 + (29*n^2 - 54*n - 14)*x^4 + (17*n^2 - 34*n - 4)*x^5 + (4*n^2 - 8*n)*x^6)*P[n - 2, x] + ((4*n - 4)*x^2 + (20*n - 16)*x^3 + (40*n - 24)*x^4 + (40*n - 16)*x^5 + (20*n - 4)*x^6 + 4*n*x^7)*D[P[n - 2, x], x] + (2*n - 1 + 8*n*x + (16*n + 2)*x^2 + (20*n - 4)*x^3 + (14*n - 9)*x^4 + (4*n - 4)*x^5)*P[n - 1, x] - (4*x^2 + 16*x^3 + 24*x^4 + 16*x^5 + 4*x^6)*D[P[n - 1, x], x])/(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; v[2] = 1 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 + (5*n^2 - 6*n + 1)*x + (14*n^2 - 16*n - 7)*x^2 + (26*n^2 - 40*n - 17)*x^3 + (29*n^2 - 54*n - 14)*x^4 + (17*n^2 - 34*n - 4)*x^5 + (4*n^2 - 8*n)*x^6)*v[i - 2] + ((4*n - 4)*x^2 + (20*n - 16)*x^3 + (40*n - 24)*x^4 + (40*n - 16)*x^5 + (20*n - 4)*x^6 + 4*n*x^7)*deriv(v[i - 2], x) + (2*n - 1 + 8*n*x + (16*n + 2)*x^2 + (20*n - 4)*x^3 + (14*n - 9)*x^4 + (4*n - 4)*x^5)*v[i - 1] - (4*x^2 + 16*x^3 + 24*x^4 + 16*x^5 + 4*x^6)*deriv(v[i - 1], 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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.