Real-rooted atlas

A038231 → Veronese 2 (j=1) → D+1

nonnegative coefficients

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

Recurrence

\(\left(-408 + 312\,x - 16\,x^{2} + 428\,n - 328\,n x + 16\,n x^{2} - 24\,n^{2} + 16\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-72\,x + 148\,x^{2} - 8\,x^{3} + 1568\,n - 1202\,n x - 18\,n x^{2} + 4\,n x^{3} - 48\,n^{2} + 57\,n^{2} x + 4\,n^{2} x^{2} + 16\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(1088\,x - 2308\,x^{2} + 268\,x^{3} - 8\,x^{4} - 96\,n x + 134\,n x^{2} - 8\,n x^{3} + 32\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
00
111
288
349150
427216288
5144016211603
674241328248776
7376329632339147604

All 15 rows

Technical details
Node id
n-20d53f5109b6258f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-652dd0e3df5ca137

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:      1;
  2:      8;
  3:     49,      1;
  4:    272,     16;
  5:   1440,    162,     1;
  6:   7424,   1328,    24;
  7:  37632,   9632,   339,   1;
  8: 188416,  64512,  3680,  32;
  9: 933888, 408576, 33984, 580, 1;
%F Formula
The row polynomials satisfy (4*n^3-24*n^2+428*n-408+(16*n^2-328*n+312)*x+(16*n-16)*x^2)*P(n,x) = (16*n^3-48*n^2+1568*n+(n^3+57*n^2-1202*n-72)*x+(4*n^2-18*n+148)*x^2+(4*n-8)*x^3)*P(n-1,x) + ((32*n^2-96*n+1088)*x-(2*n^2-134*n+2308)*x^2-(8*n-268)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 48*n^2 + 1568*n + (n^3 + 57*n^2 - 1202*n - 72)*x + (4*n^2 - 18*n + 148)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + ((32*n^2 - 96*n + 1088)*x - (2*n^2 - 134*n + 2308)*x^2 - (8*n - 268)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(4*n^3 - 24*n^2 + 428*n - 408 + (16*n^2 - 328*n + 312)*x + (16*n - 16)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^3 - 48*n^2 + 1568*n + (n^3 + 57*n^2 - 1202*n - 72)*x + (4*n^2 - 18*n + 148)*x^2 + (4*n - 8)*x^3)*v[i - 1] + ((32*n^2 - 96*n + 1088)*x - (2*n^2 - 134*n + 2308)*x^2 - (8*n - 268)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(4*n^3 - 24*n^2 + 428*n - 408 + (16*n^2 - 328*n + 312)*x + (16*n - 16)*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.