Real-rooted atlas

A188440 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(324 + 189\,x + 9\,x^{2} - 18\,x^{3} + 270\,n x + 54\,n x^{2} - 9\,n x^{3} + 126\,n^{2} x^{2} + 33\,n^{2} x^{3} + n^{2} x^{4} + 18\,n^{3} x^{3} + 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(-324 - 2025\,x - 1485\,x^{2} - 474\,x^{3} - 60\,x^{4} - 8\,x^{5} + 648\,n + 1566\,n x - 522\,n x^{2} - 609\,n x^{3} - 186\,n x^{4} - 16\,n x^{5} + 918\,n^{2} x + 1836\,n^{2} x^{2} + 465\,n^{2} x^{3} + 23\,n^{2} x^{4} - 2\,n^{2} x^{5} + 360\,n^{3} x^{2} + 594\,n^{3} x^{3} + 162\,n^{3} x^{4} + 14\,n^{3} x^{5} + 48\,n^{4} x^{3} + 63\,n^{4} x^{4} + 10\,n^{4} x^{5} + 2\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-486 - 2106\,x - 2754\,x^{2} - 1764\,x^{3} - 660\,x^{4} - 144\,x^{5} - 16\,x^{6} - 972\,n x - 3348\,n x^{2} - 3672\,n x^{3} - 1842\,n x^{4} - 468\,n x^{5} - 48\,n x^{6} - 594\,n^{2} x^{2} - 1566\,n^{2} x^{3} - 1308\,n^{2} x^{4} - 430\,n^{2} x^{5} - 52\,n^{2} x^{6} - 108\,n^{3} x^{3} - 228\,n^{3} x^{4} - 140\,n^{3} x^{5} - 24\,n^{3} x^{6} - 6\,n^{4} x^{4} - 10\,n^{4} x^{5} - 4\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
23811
33811
49483693
59483693
627216324128695
727216324128695

All 15 rows

Technical details
Node id
n-0ace05ede77c79d0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4611b6d868b2cab4

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:  3,   8;
  3:  3,   8;
  4:  9,  48,   36;
  5:  9,  48,   36;
  6: 27, 216,  324,  128;
  7: 27, 216,  324,  128;
  8: 81, 864, 1944, 1536, 400;
  9: 81, 864, 1944, 1536, 400;
%F Formula
The row polynomials satisfy (324+(270*n+189)*x+(126*n^2+54*n+9)*x^2+(18*n^3+33*n^2-9*n-18)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (648*n-324+(918*n^2+1566*n-2025)*x+(360*n^3+1836*n^2-522*n-1485)*x^2+(48*n^4+594*n^3+465*n^2-609*n-474)*x^3+(2*n^5+63*n^4+162*n^3+23*n^2-186*n-60)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (486+(972*n+2106)*x+(594*n^2+3348*n+2754)*x^2+(108*n^3+1566*n^2+3672*n+1764)*x^3+(6*n^4+228*n^3+1308*n^2+1842*n+660)*x^4+(10*n^4+140*n^3+430*n^2+468*n+144)*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[((648*n - 324 + (918*n^2 + 1566*n - 2025)*x + (360*n^3 + 1836*n^2 - 522*n - 1485)*x^2 + (48*n^4 + 594*n^3 + 465*n^2 - 609*n - 474)*x^3 + (2*n^5 + 63*n^4 + 162*n^3 + 23*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (486 + (972*n + 2106)*x + (594*n^2 + 3348*n + 2754)*x^2 + (108*n^3 + 1566*n^2 + 3672*n + 1764)*x^3 + (6*n^4 + 228*n^3 + 1308*n^2 + 1842*n + 660)*x^4 + (10*n^4 + 140*n^3 + 430*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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] = ((648*n - 324 + (918*n^2 + 1566*n - 2025)*x + (360*n^3 + 1836*n^2 - 522*n - 1485)*x^2 + (48*n^4 + 594*n^3 + 465*n^2 - 609*n - 474)*x^3 + (2*n^5 + 63*n^4 + 162*n^3 + 23*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (486 + (972*n + 2106)*x + (594*n^2 + 3348*n + 2754)*x^2 + (108*n^3 + 1566*n^2 + 3672*n + 1764)*x^3 + (6*n^4 + 228*n^3 + 1308*n^2 + 1842*n + 660)*x^4 + (10*n^4 + 140*n^3 + 430*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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

Same fingerprint ?

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