Real-rooted atlas

A178821 → Veronese 2 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-8 - 4\,x + 6\,n + 3\,n x + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-8\,x - 8\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3Pn(1)
011
155
2301545
3140105245
4560560701190
5201625206305166
6672010080378021020790
7211203696018480231078870

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(-\frac{1}{2}\,n + \frac{1}{2}\,n^{2}\right)\,P_{n}(x) = \left(6\,x + \frac{7}{2}\,n x + \frac{1}{2}\,n^{2} x\right)\,P_{n-2}(x) + \left(-4 + 3\,n + n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5\)
Technical details
Node id
n-fa2ea18f13501719 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-2d1ac866995adfa8)
Fingerprint
f6-702b8764f2c773f5

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:      5;
  2:     30,     15;
  3:    140,    105;
  4:    560,    560,     70;
  5:   2016,   2520,    630;
  6:   6720,  10080,   3780,   210;
  7:  21120,  36960,  18480,  2310;
  8:  63360, 126720,  79200, 15840,  495;
  9: 183040, 411840, 308880, 85800, 6435;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = (2*n^2+6*n-8+(n^2+3*n-4)*x)*P(n-1,x) - ((2*n+8)*x+(2*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 6*n - 8 + (n^2 + 3*n - 4)*x)*P[n - 1, x] - ((2*n + 8)*x + (2*n + 8)*x^2)*D[P[n - 1, x], x])/(n^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] = 5; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 6*n - 8 + (n^2 + 3*n - 4)*x)*v[i - 1] - ((2*n + 8)*x + (2*n + 8)*x^2)*deriv(v[i - 1], x))/(n^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.