Real-rooted atlas

A075501 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 7\,x - 5\,x^{2} - 6\,x^{3} - 6\,n x - 12\,n x^{2} - 6\,n x^{3}\right)\,P_{n-1}(x) + \left(6\,x + 18\,x^{2} + 18\,x^{3} + 6\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 36 + 90\,x + 55\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0369055181
121690011885052809
2129684241839616764549750377
3777679056257904379476262674698231056709
4466567698243539376746798481711364526766100740725529149
52799367884864489421441384400162124649441837914488458720816157905692548465
61679616848206086878494082500341696504412401660559760164323079944169980876028421409720681894161
710077696948983040984087014444752295232114059433312178313753808175573925568106598730456365623785905432922775672093370621

All 15 rows

Technical details
Node id
n-e4f7b836c7f2dd88 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f145e7ba99e5a3dd

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:        36,           90,            55;
  1:       216,          900,          1188,            505;
  2:      1296,         8424,         18396,          16764,           5497;
  3:      7776,        79056,        257904,         379476,         262674,           69823;
  4:     46656,       769824,       3539376,        7467984,        8171136,         4526766,         1007407;
  5:    279936,      7884864,      48942144,      138440016,      212464944,       183791448,        84587208,        16157905;
  6:   1679616,     84820608,     687849408,     2500341696,     5044124016,      6055976016,      4323079944,      1699808760,       284214097;
  7:  10077696,    948983040,    9840870144,    44752295232,   114059433312,    178313753808,    175573925568,    106598730456,     36562378590,     5432922775;
  8:  60466176,  10914144768,  143377060608,   802163886336,  2516191205760,   4922271942048,   6270400674000,   5226557109696,   2758190810508,   838347151938,   112034017735;
  9: 362797056, 127765029888, 2129679184896, 14504634696960, 54966851195904, 130929950658816, 207229848494976, 222344195809968, 160475957992008, 74825005993596, 20405048801004, 2476196276617;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6-(6*n-7)*x-(12*n+5)*x^2-(6*n+6)*x^3)*P(n-1,x) + (6*x+18*x^2+18*x^3+6*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 36 + 90*x + 55*x^2.
%t Mathematica
P[0, x_] := 36 + 90*x + 55*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 - (6*n - 7)*x - (12*n + 5)*x^2 - (6*n + 6)*x^3)*P[n - 1, x] + (6*x + 18*x^2 + 18*x^3 + 6*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] = 36 + 90*x + 55*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 - (6*n - 7)*x - (12*n + 5)*x^2 - (6*n + 6)*x^3)*v[i - 1] + (6*x + 18*x^2 + 18*x^3 + 6*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)

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.