Real-rooted atlas

A099040 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x + 12\,n^{2} - 18\,n^{2} x + 13\,n^{3} - 24\,n^{3} x + 4\,n^{3} x^{2} + 6\,n^{4} - 6\,n^{4} x + n^{5}\right)\,P_{n}(x) = \left(8 + 72\,x - 144\,x^{2} + 16\,x^{3} + 16\,n + 176\,n x - 312\,n x^{2} + 32\,n x^{3} + 2\,n^{2} + 200\,n^{2} x - 204\,n^{2} x^{2} + 16\,n^{2} x^{3} - 14\,n^{3} + 108\,n^{3} x - 44\,n^{3} x^{2} - 10\,n^{4} + 20\,n^{4} x - 2\,n^{5}\right)\,P_{n-1}(x) + \left(16\,x - 72\,x^{2} - 24\,x^{3} + 8\,x^{4} + 48\,n x - 152\,n x^{2} - 56\,n x^{3} + 16\,n x^{4} + 52\,n^{2} x - 94\,n^{2} x^{2} - 44\,n^{2} x^{3} + 8\,n^{2} x^{4} + 24\,n^{3} x - 12\,n^{3} x^{2} - 12\,n^{3} x^{3} + 4\,n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
101616
2016108124
300216512728
400108153620003644
500015368000691216448
600051212000345602195269024
7000080006912013171265536274368
800002000691203292804587521866241045776
90000034560439040137625614929925120003854848
10000006912329280229376052254724608000136294413826368
110000001317122293760104509441843200013629440353894448476800
12000000219521376256130636804300800061332480389283848998912166729664
130000000458752104509446451200016355328019464192010798694422478848564082688
14000000065536522547264512000286218240583925760593928192292225024552960001881396224
Technical details
Node id
n-b49109a87841a445 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6b8c9911c47890c9

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: 0, 16;
  2: 0, 16, 108;
  3: 0,  0, 216,  512;
  4: 0,  0, 108, 1536,  2000;
  5: 0,  0,   0, 1536,  8000,  6912;
  6: 0,  0,   0,  512, 12000, 34560,  21952;
  7: 0,  0,   0,    0,  8000, 69120, 131712,   65536;
  8: 0,  0,   0,    0,  2000, 69120, 329280,  458752,  186624;
  9: 0,  0,   0,    0,     0, 34560, 439040, 1376256, 1492992, 512000;
%F Formula
The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(6*n^4+24*n^3+18*n^2-4*n)*x+4*n^3*x^2)*P(n,x) = -(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8-(20*n^4+108*n^3+200*n^2+176*n+72)*x+(44*n^3+204*n^2+312*n+144)*x^2-(16*n^2+32*n+16)*x^3)*P(n-1,x) + ((4*n^4+24*n^3+52*n^2+48*n+16)*x+(2*n^4-12*n^3-94*n^2-152*n-72)*x^2-(12*n^3+44*n^2+56*n+24)*x^3+(8*n^2+16*n+8)*x^4)*d/dx 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[(-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (20*n^4 + 108*n^3 + 200*n^2 + 176*n + 72)*x + (44*n^3 + 204*n^2 + 312*n + 144)*x^2 - (16*n^2 + 32*n + 16)*x^3)*P[n - 1, x] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (2*n^4 - 12*n^3 - 94*n^2 - 152*n - 72)*x^2 - (12*n^3 + 44*n^2 + 56*n + 24)*x^3 + (8*n^2 + 16*n + 8)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + 4*n^3*x^2)]];
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] = (-(2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8 - (20*n^4 + 108*n^3 + 200*n^2 + 176*n + 72)*x + (44*n^3 + 204*n^2 + 312*n + 144)*x^2 - (16*n^2 + 32*n + 16)*x^3)*v[i - 1] + ((4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x + (2*n^4 - 12*n^3 - 94*n^2 - 152*n - 72)*x^2 - (12*n^3 + 44*n^2 + 56*n + 24)*x^3 + (8*n^2 + 16*n + 8)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + 4*n^3*x^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

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.