Real-rooted atlas

A100861 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 12\,x + 8\,x^{2} - 8\,x^{3} + 10\,n x + 10\,n x^{2} - 4\,n x^{3} + 14\,n^{2} x^{2} + 14\,n^{2} x^{3} + n^{2} x^{4} + 6\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-18\,x - 88\,x^{2} - 112\,x^{3} - 16\,x^{4} - 8\,x^{5} + 4\,n + 23\,n x + 24\,n x^{2} - 48\,n x^{3} - 80\,n x^{4} - 16\,n x^{5} + 17\,n^{2} x + 76\,n^{2} x^{2} + 88\,n^{2} x^{3} + 12\,n^{2} x^{4} - 2\,n^{2} x^{5} + 20\,n^{3} x^{2} + 72\,n^{3} x^{3} + 65\,n^{3} x^{4} + 14\,n^{3} x^{5} + 8\,n^{4} x^{3} + 22\,n^{4} x^{4} + 10\,n^{4} x^{5} + n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1 - 12\,x - 52\,x^{2} - 104\,x^{3} - 100\,x^{4} - 48\,x^{5} - 16\,x^{6} - 6\,n x - 49\,n x^{2} - 156\,n x^{3} - 240\,n x^{4} - 176\,n x^{5} - 48\,n x^{6} - 11\,n^{2} x^{2} - 70\,n^{2} x^{3} - 161\,n^{2} x^{4} - 156\,n^{2} x^{5} - 52\,n^{2} x^{6} - 6\,n^{3} x^{3} - 30\,n^{3} x^{4} - 48\,n^{3} x^{5} - 24\,n^{3} x^{6} - n^{4} x^{4} - 4\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
22810
342428
41096108214
526320540886
6761200324019206436
72324368151201344033160

All 15 rows

Technical details
Node id
n-677c568a2dd425e4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-bcf4ade9a6a32b10

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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,     8;
  3:    4,    24;
  4:   10,    96,    108;
  5:   26,   320,    540;
  6:   76,  1200,   3240,   1920;
  7:  232,  4368,  15120,  13440;
  8:  764, 17024,  75600, 107520,  42000;
  9: 2620, 66816, 353808, 645120, 378000;
%F Formula
The row polynomials satisfy (4+(10*n+12)*x+(14*n^2+10*n+8)*x^2+(6*n^3+14*n^2-4*n-8)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (4*n+(17*n^2+23*n-18)*x+(20*n^3+76*n^2+24*n-88)*x^2+(8*n^4+72*n^3+88*n^2-48*n-112)*x^3+(n^5+22*n^4+65*n^3+12*n^2-80*n-16)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (1+(6*n+12)*x+(11*n^2+49*n+52)*x^2+(6*n^3+70*n^2+156*n+104)*x^3+(n^4+30*n^3+161*n^2+240*n+100)*x^4+(4*n^4+48*n^3+156*n^2+176*n+48)*x^5+(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*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[((4*n + (17*n^2 + 23*n - 18)*x + (20*n^3 + 76*n^2 + 24*n - 88)*x^2 + (8*n^4 + 72*n^3 + 88*n^2 - 48*n - 112)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (1 + (6*n + 12)*x + (11*n^2 + 49*n + 52)*x^2 + (6*n^3 + 70*n^2 + 156*n + 104)*x^3 + (n^4 + 30*n^3 + 161*n^2 + 240*n + 100)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 12)*x + (14*n^2 + 10*n + 8)*x^2 + (6*n^3 + 14*n^2 - 4*n - 8)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];
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] = ((4*n + (17*n^2 + 23*n - 18)*x + (20*n^3 + 76*n^2 + 24*n - 88)*x^2 + (8*n^4 + 72*n^3 + 88*n^2 - 48*n - 112)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (1 + (6*n + 12)*x + (11*n^2 + 49*n + 52)*x^2 + (6*n^3 + 70*n^2 + 156*n + 104)*x^3 + (n^4 + 30*n^3 + 161*n^2 + 240*n + 100)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 12)*x + (14*n^2 + 10*n + 8)*x^2 + (6*n^3 + 14*n^2 - 4*n - 8)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.