Real-rooted atlas

A209599 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-25 + 190\,x - 69\,x^{2} + 2\,x^{3} - 8\,x^{4} - 75\,n x + 285\,n x^{2} - 22\,n x^{3} - 5\,n x^{4} - 60\,n^{2} x^{2} + 125\,n^{2} x^{3} + 15\,n^{2} x^{4} + n^{2} x^{5} - 15\,n^{3} x^{3} + 19\,n^{3} x^{4} + 2\,n^{3} x^{5} - n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n}(x) = \left(-25 + 90\,x + 271\,x^{2} + 246\,x^{3} + 110\,x^{4} + 24\,x^{5} + 4\,x^{6} - 75\,n x + 90\,n x^{2} + 348\,n x^{3} + 261\,n x^{4} + 90\,n x^{5} + 12\,n x^{6} - 60\,n^{2} x^{2} + 20\,n^{2} x^{3} + 146\,n^{2} x^{4} + 79\,n^{2} x^{5} + 13\,n^{2} x^{6} - 15\,n^{3} x^{3} + 21\,n^{3} x^{5} + 6\,n^{3} x^{6} - n^{4} x^{4} + n^{4} x^{6}\right)\,P_{n-2}(x) + \left(-25 + 190\,x + 106\,x^{2} + 52\,x^{3} - 15\,x^{4} - 2\,x^{5} - 75\,n x + 250\,n x^{2} + 103\,n x^{3} + 31\,n x^{4} - 60\,n^{2} x^{2} + 110\,n^{2} x^{3} + 44\,n^{2} x^{4} + 5\,n^{2} x^{5} - 15\,n^{3} x^{3} + 17\,n^{3} x^{4} + 4\,n^{3} x^{5} - n^{4} x^{4} + n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
2347
351217
4828945
5136036109
62112010816265
73423227980625
855436657288251461
98980414588641503365
10144146030962320625367681
11233262063635776212525217369
1237746561274413600637511884938989
136108208250023068817525450039286925
149871437248222669444515014868205864192665
Technical details
Node id
n-defc18c47aef3671 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1c1fb59272f650a5

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:  2;
  2:  3,   4;
  3:  5,  12;
  4:  8,  28,    9;
  5: 13,  60,   36;
  6: 21, 120,  108,  16;
  7: 34, 232,  279,  80;
  8: 55, 436,  657, 288,  25;
  9: 89, 804, 1458, 864, 150;
%F Formula
The row polynomials satisfy -(25+(75*n-190)*x+(60*n^2-285*n+69)*x^2+(15*n^3-125*n^2+22*n-2)*x^3+(n^4-19*n^3-15*n^2+5*n+8)*x^4-(n^4+2*n^3+n^2)*x^5)*P(n,x) = -(25+(75*n-90)*x+(60*n^2-90*n-271)*x^2+(15*n^3-20*n^2-348*n-246)*x^3+(n^4-146*n^2-261*n-110)*x^4-(21*n^3+79*n^2+90*n+24)*x^5-(n^4+6*n^3+13*n^2+12*n+4)*x^6)*P(n-2,x) - (25+(75*n-190)*x+(60*n^2-250*n-106)*x^2+(15*n^3-110*n^2-103*n-52)*x^3+(n^4-17*n^3-44*n^2-31*n+15)*x^4-(n^4+4*n^3+5*n^2-2)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(25 + (75*n - 90)*x + (60*n^2 - 90*n - 271)*x^2 + (15*n^3 - 20*n^2 - 348*n - 246)*x^3 + (n^4 - 146*n^2 - 261*n - 110)*x^4 - (21*n^3 + 79*n^2 + 90*n + 24)*x^5 - (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^6)*P[n - 2, x] - (25 + (75*n - 190)*x + (60*n^2 - 250*n - 106)*x^2 + (15*n^3 - 110*n^2 - 103*n - 52)*x^3 + (n^4 - 17*n^3 - 44*n^2 - 31*n + 15)*x^4 - (n^4 + 4*n^3 + 5*n^2 - 2)*x^5)*P[n - 1, x])/(-25 - (75*n - 190)*x - (60*n^2 - 285*n + 69)*x^2 - (15*n^3 - 125*n^2 + 22*n - 2)*x^3 - (n^4 - 19*n^3 - 15*n^2 + 5*n + 8)*x^4 + (n^4 + 2*n^3 + n^2)*x^5)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + (75*n - 90)*x + (60*n^2 - 90*n - 271)*x^2 + (15*n^3 - 20*n^2 - 348*n - 246)*x^3 + (n^4 - 146*n^2 - 261*n - 110)*x^4 - (21*n^3 + 79*n^2 + 90*n + 24)*x^5 - (n^4 + 6*n^3 + 13*n^2 + 12*n + 4)*x^6)*v[i - 2] - (25 + (75*n - 190)*x + (60*n^2 - 250*n - 106)*x^2 + (15*n^3 - 110*n^2 - 103*n - 52)*x^3 + (n^4 - 17*n^3 - 44*n^2 - 31*n + 15)*x^4 - (n^4 + 4*n^3 + 5*n^2 - 2)*x^5)*v[i - 1])/(-25 - (75*n - 190)*x - (60*n^2 - 285*n + 69)*x^2 - (15*n^3 - 125*n^2 + 22*n - 2)*x^3 - (n^4 - 19*n^3 - 15*n^2 + 5*n + 8)*x^4 + (n^4 + 2*n^3 + n^2)*x^5)); 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.