Real-rooted atlas

A216973 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 11\,x - 11\,x^{2} + x^{3} + n - 18\,n x + 27\,n x^{2} - 4\,n x^{3} + 7\,n^{2} x - 22\,n^{2} x^{2} + 6\,n^{2} x^{3} + 6\,n^{3} x^{2} - 4\,n^{3} x^{3} + n^{4} x^{3}\right)\,P_{n}(x) = \left(n - 3\,n x - 3\,n x^{2} + n x^{3} + 7\,n^{2} x + 3\,n^{2} x^{2} - 4\,n^{2} x^{3} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3} + n^{4} x^{3} + n^{4} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
221618
334881132
4496324256680
5516081012806252880
66240162038403750129610752
7733628358960131259072240136736
884484536179203500036288192084096117504
995766804322567875010886486436368646561357120
101072097205376015750027216028812018432065610100001041920
11118801336584480288750598752792330675840360855110000146412939904
12121056178201267204950001197504190159220275201443420660000175692207368067072
13131248231661830408043752223936412011652715524691115286000011419982695682856121618688
1414145629484256256125125038918888240232123002881313512210010000532932418869763998543841656770560
Technical details
Node id
n-4e8197f454564049 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-eef2158e879e5e15

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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: 0;
  1: 1;
  2: 2,  16;
  3: 3,  48,   81;
  4: 4,  96,  324,   256;
  5: 5, 160,  810,  1280,   625;
  6: 6, 240, 1620,  3840,  3750,   1296;
  7: 7, 336, 2835,  8960, 13125,   9072,  2401;
  8: 8, 448, 4536, 17920, 35000,  36288, 19208,  4096;
  9: 9, 576, 6804, 32256, 78750, 108864, 86436, 36864, 6561;
%F Formula
The row polynomials satisfy (n-1+(7*n^2-18*n+11)*x+(6*n^3-22*n^2+27*n-11)*x^2+(n^4-4*n^3+6*n^2-4*n+1)*x^3)*P(n,x) = (n+(7*n^2-3*n)*x+(6*n^3+3*n^2-3*n)*x^2+(n^4+6*n^3-4*n^2+n)*x^3+n^4*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (7*n^2 - 3*n)*x + (6*n^3 + 3*n^2 - 3*n)*x^2 + (n^4 + 6*n^3 - 4*n^2 + n)*x^3 + n^4*x^4)*P[n - 1, x])/(n - 1 + (7*n^2 - 18*n + 11)*x + (6*n^3 - 22*n^2 + 27*n - 11)*x^2 + (n^4 - 4*n^3 + 6*n^2 - 4*n + 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + (7*n^2 - 3*n)*x + (6*n^3 + 3*n^2 - 3*n)*x^2 + (n^4 + 6*n^3 - 4*n^2 + n)*x^3 + n^4*x^4)*v[i - 1])/(n - 1 + (7*n^2 - 18*n + 11)*x + (6*n^3 - 22*n^2 + 27*n - 11)*x^2 + (n^4 - 4*n^3 + 6*n^2 - 4*n + 1)*x^3)); 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.