Real-rooted atlas

A159854 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 + 6\,x + 6\,x^{2} - 4\,x^{3} - 6\,n - 7\,n x - 10\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(6 + 30\,x + 6\,x^{2} - 14\,x^{3} + 4\,x^{4} - 6\,n - 19\,n x - 29\,n x^{2} + 6\,n x^{3} + 4\,n x^{4} + 6\,n^{2} x^{2} + 4\,n^{2} x^{3} + n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 4\,x^{2} + 20\,x^{3} + 4\,x^{4} - 8\,x^{5} + 6\,n x^{2} + 2\,n x^{3} - 8\,n x^{4} + 2\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + x\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4Pn(1)
0112
1112
21315
317311
41139124
512125552
613160191112
7143126637240

All 15 rows

Technical details
Node id
n-ae0bf4fdd6c3bd26 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-32d32fad1924fa68

Observed on the stored rows ?

  • Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
  • 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: 1,  1;
  2: 1,  3,   1;
  3: 1,  7,   3;
  4: 1, 13,   9,   1;
  5: 1, 21,  25,   5;
  6: 1, 31,  60,  19,   1;
  7: 1, 43, 126,  63,   7;
  8: 1, 57, 238, 182,  33, 1;
  9: 1, 73, 414, 462, 129, 9;
%F Formula
The row polynomials satisfy -(6*n-6-(n^3-7*n+6)*x-(4*n^2-10*n+6)*x^2-(4*n-4)*x^3)*P(n,x) = -(6*n-6-(n^3-19*n+30)*x-(n^3+6*n^2-29*n+6)*x^2-(4*n^2+6*n-14)*x^3-(4*n+4)*x^4)*P(n-1,x) - (12*x-(2*n^2+6*n-4)*x^2+(2*n^2-2*n-20)*x^3+(8*n-4)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + x, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1 + x;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (n^3 - 19*n + 30)*x - (n^3 + 6*n^2 - 29*n + 6)*x^2 - (4*n^2 + 6*n - 14)*x^3 - (4*n + 4)*x^4)*P[n - 1, x] - (12*x - (2*n^2 + 6*n - 4)*x^2 + (2*n^2 - 2*n - 20)*x^3 + (8*n - 4)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - 7*n + 6)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (n^3 - 19*n + 30)*x - (n^3 + 6*n^2 - 29*n + 6)*x^2 - (4*n^2 + 6*n - 14)*x^3 - (4*n + 4)*x^4)*v[i - 1] - (12*x - (2*n^2 + 6*n - 4)*x^2 + (2*n^2 - 2*n - 20)*x^3 + (8*n - 4)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - 7*n + 6)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.