Real-rooted atlas

A013609 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 22\,x - 8\,x^{2} + 11\,n - 30\,n x + 8\,n x^{2} - 8\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-1 + 26\,x + 48\,x^{2} - 32\,x^{3} + 3\,n - 10\,n x - 80\,n x^{2} + 32\,n x^{3} - 4\,n^{2} - 24\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} + 64\,x^{3} - 64\,x^{4} - 4\,n x + 32\,n x^{2} - 64\,n x^{3} + 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2549
3131225
425561697
54120080321
661540432641097
785120419044483641

All 15 rows

Technical details
Node id
n-23f770c136ddec74 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-18a4c119033fa265

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:   5,    4;
  3:  13,   12;
  4:  25,   56,    16;
  5:  41,  200,    80;
  6:  61,  540,   432,    64;
  7:  85, 1204,  1904,   448;
  8: 113, 2352,  6496,  2816,  256;
  9: 145, 4176, 18144, 14592, 2304;
%F Formula
The row polynomials satisfy (2*n^3-8*n^2+11*n-5+(8*n^2-30*n+22)*x+(8*n-8)*x^2)*P(n,x) = (2*n^3-4*n^2+3*n-1+(8*n^3-24*n^2-10*n+26)*x+(32*n^2-80*n+48)*x^2+(32*n-32)*x^3)*P(n-1,x) + ((4*n^2-4*n-2)*x-(16*n^2-32*n+4)*x^2-(64*n-64)*x^3-64*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[((2*n^3 - 4*n^2 + 3*n - 1 + (8*n^3 - 24*n^2 - 10*n + 26)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n - 2)*x - (16*n^2 - 32*n + 4)*x^2 - (64*n - 64)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 8*n^2 + 11*n - 5 + (8*n^2 - 30*n + 22)*x + (8*n - 8)*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] = ((2*n^3 - 4*n^2 + 3*n - 1 + (8*n^3 - 24*n^2 - 10*n + 26)*x + (32*n^2 - 80*n + 48)*x^2 + (32*n - 32)*x^3)*v[i - 1] + ((4*n^2 - 4*n - 2)*x - (16*n^2 - 32*n + 4)*x^2 - (64*n - 64)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 8*n^2 + 11*n - 5 + (8*n^2 - 30*n + 22)*x + (8*n - 8)*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.