Real-rooted atlas

A027465 → Veronese 2 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-60 + 138\,x - 12\,x^{2} + 69\,n - 150\,n x + 12\,n x^{2} - 12\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(-162 + 432\,x + 6\,x^{2} - 4\,x^{3} + 171\,n - 465\,n x - 10\,n x^{2} + 4\,n x^{3} - 18\,n^{2} + 32\,n^{2} x + 4\,n^{2} x^{2} + 9\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-324\,x - 648\,x^{2} + 148\,x^{3} - 8\,x^{4} - 18\,n x + 74\,n x^{2} - 8\,n x^{3} + 18\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3Pn(1)
011
133
210111
336945
4135561192
551330015828
61944148513813568
77290699310082115312

All 15 rows

Technical details
Node id
n-89ac50958b396ed9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4c5c7f89b44291d6

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:     3;
  2:    10,      1;
  3:    36,      9;
  4:   135,     56,     1;
  5:   513,    300,    15;
  6:  1944,   1485,   138,    1;
  7:  7290,   6993,  1008,   21;
  8: 26973,  31752,  6426,  256,  1;
  9: 98415, 139968, 37422, 2376, 27;
%F Formula
The row polynomials satisfy (3*n^3-12*n^2+69*n-60+(12*n^2-150*n+138)*x+(12*n-12)*x^2)*P(n,x) = (9*n^3-18*n^2+171*n-162+(n^3+32*n^2-465*n+432)*x+(4*n^2-10*n+6)*x^2+(4*n-4)*x^3)*P(n-1,x) + ((18*n^2-18*n-324)*x-(2*n^2-74*n+648)*x^2-(8*n-148)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 18*n^2 + 171*n - 162 + (n^3 + 32*n^2 - 465*n + 432)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + ((18*n^2 - 18*n - 324)*x - (2*n^2 - 74*n + 648)*x^2 - (8*n - 148)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 69*n - 60 + (12*n^2 - 150*n + 138)*x + (12*n - 12)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 18*n^2 + 171*n - 162 + (n^3 + 32*n^2 - 465*n + 432)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)*v[i - 1] + ((18*n^2 - 18*n - 324)*x - (2*n^2 - 74*n + 648)*x^2 - (8*n - 148)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 69*n - 60 + (12*n^2 - 150*n + 138)*x + (12*n - 12)*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.