Real-rooted atlas

A008277 → (θ+1)² → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(-10\,x - 37\,x^{2} - 9\,x^{3} + 131\,x^{4} + 227\,x^{5} + 150\,x^{6} + 36\,x^{7} + 2\,n x + 13\,n x^{2} + 7\,n x^{3} - 58\,n x^{4} - 116\,n x^{5} - 83\,n x^{6} - 21\,n x^{7} - n^{2} x^{2} - n^{2} x^{3} + 6\,n^{2} x^{4} + 14\,n^{2} x^{5} + 11\,n^{2} x^{6} + 3\,n^{2} x^{7}\right)\,P_{n-2}(x) + \left(-2\,x - 16\,x^{2} - 22\,x^{3} + 56\,x^{4} + 194\,x^{5} + 224\,x^{6} + 118\,x^{7} + 24\,x^{8} + 2\,n x^{2} + 4\,n x^{3} - 10\,n x^{4} - 40\,n x^{5} - 50\,n x^{6} - 28\,n x^{7} - 6\,n x^{8}\right)\,P_{n-2}'(x) + \left(-x^{2} - 3\,x^{3} + 3\,x^{4} + 25\,x^{5} + 45\,x^{6} + 39\,x^{7} + 17\,x^{8} + 3\,x^{9}\right)\,P_{n-2}''(x) + \left(1 + 12\,x + 26\,x^{2} + 16\,x^{3} - 3\,x^{4} - 4\,x^{5} - 2\,n x - 4\,n x^{2} + 4\,n x^{4} + 2\,n x^{5}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 4\,x^{3} - 4\,x^{4} - 6\,x^{5} - 2\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2156
31142237
413111399244
51644117944711741
61129131642245386238613442
71258398418976390893665812867111833
81515117637834123224934620325466373681997416

All 15 rows

Technical details
Node id
n-a12656369f6936b4 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ab2f87d5d6cfe794

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,    5;
   3: 1,   14,     22;
   4: 1,   31,    113,      99;
   5: 1,   64,    411,     794,     471;
   6: 1,  129,   1316,    4224,    5386,     2386;
   7: 1,  258,   3984,   18976,   39089,    36658,    12867;
   8: 1,  515,  11763,   78341,  232249,   346203,   254663,    73681;
   9: 1, 1028,  34393,  309146, 1239520,  2613812,  3024079,  1820546,   446620;
  10: 1, 2053, 100358, 1190786, 6215165, 17404753, 28173556, 26480080, 13444949, 2856457;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = ((2*n-10)*x-(n^2-13*n+37)*x^2-(n^2-7*n+9)*x^3+(6*n^2-58*n+131)*x^4+(14*n^2-116*n+227)*x^5+(11*n^2-83*n+150)*x^6+(3*n^2-21*n+36)*x^7)*P(n-2,x) - (2*x-(2*n-16)*x^2-(4*n-22)*x^3+(10*n-56)*x^4+(40*n-194)*x^5+(50*n-224)*x^6+(28*n-118)*x^7+(6*n-24)*x^8)*d/dx P(n-2,x) - (x^2+3*x^3-3*x^4-25*x^5-45*x^6-39*x^7-17*x^8-3*x^9)*d^2/dx^2 P(n-2,x) + (1-(2*n-12)*x-(4*n-26)*x^2+16*x^3+(4*n-3)*x^4+(2*n-4)*x^5)*P(n-1,x) + (2*x+6*x^2+4*x^3-4*x^4-6*x^5-2*x^6)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 5*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 10)*x - (n^2 - 13*n + 37)*x^2 - (n^2 - 7*n + 9)*x^3 + (6*n^2 - 58*n + 131)*x^4 + (14*n^2 - 116*n + 227)*x^5 + (11*n^2 - 83*n + 150)*x^6 + (3*n^2 - 21*n + 36)*x^7)*P[n - 2, x] - (2*x - (2*n - 16)*x^2 - (4*n - 22)*x^3 + (10*n - 56)*x^4 + (40*n - 194)*x^5 + (50*n - 224)*x^6 + (28*n - 118)*x^7 + (6*n - 24)*x^8)*D[P[n - 2, x], x] - (x^2 + 3*x^3 - 3*x^4 - 25*x^5 - 45*x^6 - 39*x^7 - 17*x^8 - 3*x^9)*D[P[n - 2, x], {x, 2}] + (1 - (2*n - 12)*x - (4*n - 26)*x^2 + 16*x^3 + (4*n - 3)*x^4 + (2*n - 4)*x^5)*P[n - 1, x] + (2*x + 6*x^2 + 4*x^3 - 4*x^4 - 6*x^5 - 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 5*x; for(n = 3, N, my(i = n); v[i] = (((2*n - 10)*x - (n^2 - 13*n + 37)*x^2 - (n^2 - 7*n + 9)*x^3 + (6*n^2 - 58*n + 131)*x^4 + (14*n^2 - 116*n + 227)*x^5 + (11*n^2 - 83*n + 150)*x^6 + (3*n^2 - 21*n + 36)*x^7)*v[i - 2] - (2*x - (2*n - 16)*x^2 - (4*n - 22)*x^3 + (10*n - 56)*x^4 + (40*n - 194)*x^5 + (50*n - 224)*x^6 + (28*n - 118)*x^7 + (6*n - 24)*x^8)*deriv(v[i - 2], x) - (x^2 + 3*x^3 - 3*x^4 - 25*x^5 - 45*x^6 - 39*x^7 - 17*x^8 - 3*x^9)*deriv(deriv(v[i - 2], x), x) + (1 - (2*n - 12)*x - (4*n - 26)*x^2 + 16*x^3 + (4*n - 3)*x^4 + (2*n - 4)*x^5)*v[i - 1] + (2*x + 6*x^2 + 4*x^3 - 4*x^4 - 6*x^5 - 2*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, v[i])))};
T(10)

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