Real-rooted atlas

A038208 → Veronese 2 (j=1) → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,x + 4\,n - 22\,n x + 28\,n^{2} x - n^{2} x^{2} - 10\,n^{3} x + 3\,n^{3} x^{2} - 3\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-64\,x - 32\,x^{2} + 8\,n + 24\,n x + 24\,n x^{2} + 36\,n^{2} x + 68\,n^{2} x^{2} - 4\,n^{2} x^{3} - 20\,n^{3} x - 34\,n^{3} x^{2} - 6\,n^{3} x^{3} - 4\,n^{4} x^{2} + 2\,n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 16\,x^{3} - 8\,n x^{2} + 8\,n x^{3} - 40\,n^{2} x^{2} + 44\,n^{2} x^{3} - 4\,n^{2} x^{4} + 8\,n^{3} x^{3} - 8\,n^{3} x^{4} + 4\,n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
00
122
288
3243256
464256320
516012802881728
6384512034568960
78961792024192204845056

All 15 rows

Technical details
Node id
n-b62e3fe53a56a2b6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4b3b88e53403151e

Observed on the stored rows ?

  • Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half affine: slope 1, intercept -1)
  • 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:    0;
  1:    2;
  2:    8;
  3:   24,     32;
  4:   64,    256;
  5:  160,   1280,    288;
  6:  384,   5120,   3456;
  7:  896,  17920,  24192,   2048;
  8: 2048,  57344, 129024,  32768;
  9: 4608, 172032, 580608, 294912, 12800;
%F Formula
The row polynomials satisfy (4*n-4-(10*n^3-28*n^2+22*n-4)*x+(n^5-3*n^4+3*n^3-n^2)*x^2)*P(n,x) = (8*n-(20*n^3-36*n^2-24*n+64)*x+(2*n^5-4*n^4-34*n^3+68*n^2+24*n-32)*x^2+(2*n^5-6*n^3-4*n^2)*x^3)*P(n-1,x) + (16*x-(40*n^2+8*n)*x^2+(4*n^4+8*n^3+44*n^2+8*n-16)*x^3-(4*n^4+8*n^3+4*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - (20*n^3 - 36*n^2 - 24*n + 64)*x + (2*n^5 - 4*n^4 - 34*n^3 + 68*n^2 + 24*n - 32)*x^2 + (2*n^5 - 6*n^3 - 4*n^2)*x^3)*P[n - 1, x] + (16*x - (40*n^2 + 8*n)*x^2 + (4*n^4 + 8*n^3 + 44*n^2 + 8*n - 16)*x^3 - (4*n^4 + 8*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(4*n - 4 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (n^5 - 3*n^4 + 3*n^3 - 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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n - (20*n^3 - 36*n^2 - 24*n + 64)*x + (2*n^5 - 4*n^4 - 34*n^3 + 68*n^2 + 24*n - 32)*x^2 + (2*n^5 - 6*n^3 - 4*n^2)*x^3)*v[i - 1] + (16*x - (40*n^2 + 8*n)*x^2 + (4*n^4 + 8*n^3 + 44*n^2 + 8*n - 16)*x^3 - (4*n^4 + 8*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(4*n - 4 - (10*n^3 - 28*n^2 + 22*n - 4)*x + (n^5 - 3*n^4 + 3*n^3 - 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

Same fingerprint ?

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.