Real-rooted atlas

A127773 → (θ+1)² → reverse

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4\,n^{2} - 12\,n^{3} + 13\,n^{4} - 6\,n^{5} + n^{6}\right)\,P_{n}(x) = \left(n^{2} - 2\,n^{4} + n^{6}\right)\,P_{n-3}(x) + \left(-12\,n^{2} - 12\,n^{3} + 9\,n^{4} + 6\,n^{5} - 3\,n^{6}\right)\,P_{n-2}(x) + \left(12 - 12\,n - 21\,n^{2} + 24\,n^{3} + 6\,n^{4} - 12\,n^{5} + 3\,n^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12\)
\(P_{2}(x) = 54\)

Rows ?

nx0Pn(1)
011
11212
25454
3160160
4375375
5756756
613721372
723042304

All 15 rows

Technical details
Node id
n-c959fc01c927f962 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-edf6954a4cd8adfc

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (affine: slope 0, intercept 0)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:   12;
  2:   54;
  3:  160;
  4:  375;
  5:  756;
  6: 1372;
  7: 2304;
  8: 3645;
  9: 5500;
%F Formula
The row polynomials satisfy (n^6-6*n^5+13*n^4-12*n^3+4*n^2)*P(n,x) = (n^6-2*n^4+n^2)*P(n-3,x) - (3*n^6-6*n^5-9*n^4+12*n^3+12*n^2)*P(n-2,x) + (3*n^6-12*n^5+6*n^4+24*n^3-21*n^2-12*n+12)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 12, P(2,x) = 54.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12;
P[2, x_] := 54;
P[n_, x_] := P[n, x] = Expand[Together[((n^6 - 2*n^4 + n^2)*P[n - 3, x] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*P[n - 2, x] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*P[n - 1, x])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^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] = 12; v[3] = 54; for(n = 3, N, my(i = n + 1); v[i] = ((n^6 - 2*n^4 + n^2)*v[i - 3] - (3*n^6 - 6*n^5 - 9*n^4 + 12*n^3 + 12*n^2)*v[i - 2] + (3*n^6 - 12*n^5 + 6*n^4 + 24*n^3 - 21*n^2 - 12*n + 12)*v[i - 1])/(n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.