Real-rooted atlas

A178820 → Veronese 2 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-\frac{1}{2}\,n + \frac{1}{2}\,n^{2}\right)\,P_{n}(x) = \left(9\,x + \frac{15}{2}\,n x + \frac{3}{2}\,n^{2} x\right)\,P_{n-2}(x) + \left(-3 + 2\,n + n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3Pn(1)
011
144
2203050
380180260
42808403151435
5896336025206776
626881209613608226830660
77680403206048022680131160

All 15 rows

Technical details
Node id
n-333c4a5d7cc8fd49 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f03a94f3319f9845

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:     4;
  2:    20,     30;
  3:    80,    180;
  4:   280,    840,    315;
  5:   896,   3360,   2520;
  6:  2688,  12096,  13608,   2268;
  7:  7680,  40320,  60480,  22680;
  8: 21120, 126720, 237600, 142560,  13365;
  9: 56320, 380160, 855360, 712800, 160380;
%F Formula
The row polynomials satisfy ((1/2)*n^2-(1/2)*n)*P(n,x) = ((3/2)*n^2+(15/2)*n+9)*x*P(n-2,x) + (n^2+2*n-3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(((3/2)*n^2 + (15/2)*n + 9)*x*P[n - 2, x] + (n^2 + 2*n - 3)*P[n - 1, x])/((1/2)*n^2 - (1/2)*n)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (((3/2)*n^2 + (15/2)*n + 9)*x*v[i - 2] + (n^2 + 2*n - 3)*v[i - 1])/((1/2)*n^2 - (1/2)*n)); 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.