Real-rooted atlas

A013610 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 - 12\,x + 4\,n + 10\,n x + 24\,n^{2} x - 9\,n^{2} x^{2} - 22\,n^{3} x - 9\,n^{3} x^{2} + 9\,n^{4} x^{2} + 9\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-4 - 156\,x + 36\,x^{2} - 324\,x^{3} + 4\,n + 154\,n x + 558\,n x^{2} - 648\,n x^{3} + 24\,n^{2} x - 261\,n^{2} x^{2} - 81\,n^{2} x^{3} - 22\,n^{3} x - 351\,n^{3} x^{2} + 567\,n^{3} x^{3} + 9\,n^{4} x^{2} + 405\,n^{4} x^{3} + 9\,n^{5} x^{2} + 81\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x - 88\,x^{2} + 216\,x^{3} - 648\,x^{4} - 84\,n x^{2} + 972\,n x^{3} - 1944\,n x^{4} - 44\,n^{2} x^{2} + 630\,n^{2} x^{3} - 2106\,n^{2} x^{4} + 108\,n^{3} x^{3} - 972\,n^{3} x^{4} + 18\,n^{4} x^{3} - 162\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
213637
31108109
41216729946
5136036454006
61540109351166423140
717562551581648107920

All 15 rows

Technical details
Node id
n-c759d612d850cdda · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2bd27bb3734f2002

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: 1,   36;
  3: 1,  108;
  4: 1,  216,   729;
  5: 1,  360,  3645;
  6: 1,  540, 10935,  11664;
  7: 1,  756, 25515,  81648;
  8: 1, 1008, 51030, 326592,  164025;
  9: 1, 1296, 91854, 979776, 1476225;
%F Formula
The row polynomials satisfy (4*n-4-(22*n^3-24*n^2-10*n+12)*x+(9*n^5+9*n^4-9*n^3-9*n^2)*x^2)*P(n,x) = (4*n-4-(22*n^3-24*n^2-154*n+156)*x+(9*n^5+9*n^4-351*n^3-261*n^2+558*n+36)*x^2+(81*n^5+405*n^4+567*n^3-81*n^2-648*n-324)*x^3)*P(n-1,x) + (8*x-(44*n^2+84*n+88)*x^2+(18*n^4+108*n^3+630*n^2+972*n+216)*x^3-(162*n^4+972*n^3+2106*n^2+1944*n+648)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (9*n^5 + 9*n^4 - 351*n^3 - 261*n^2 + 558*n + 36)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*P[n - 1, x] + (8*x - (44*n^2 + 84*n + 88)*x^2 + (18*n^4 + 108*n^3 + 630*n^2 + 972*n + 216)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*D[P[n - 1, x], x])/(4*n - 4 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (9*n^5 + 9*n^4 - 9*n^3 - 9*n^2)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (9*n^5 + 9*n^4 - 351*n^3 - 261*n^2 + 558*n + 36)*x^2 + (81*n^5 + 405*n^4 + 567*n^3 - 81*n^2 - 648*n - 324)*x^3)*v[i - 1] + (8*x - (44*n^2 + 84*n + 88)*x^2 + (18*n^4 + 108*n^3 + 630*n^2 + 972*n + 216)*x^3 - (162*n^4 + 972*n^3 + 2106*n^2 + 1944*n + 648)*x^4)*deriv(v[i - 1], x))/(4*n - 4 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (9*n^5 + 9*n^4 - 9*n^3 - 9*n^2)*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.