Real-rooted atlas

A049020 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(-1 + 2\,x + 5\,x^{2} - 2\,x^{4} - n - 2\,n x + 8\,n x^{2} + 14\,n x^{3} + n x^{4} - 4\,n x^{5} - 2\,n^{2} x - 6\,n^{2} x^{2} + 10\,n^{2} x^{4} + 6\,n^{2} x^{5} - n^{3} x^{2} - 4\,n^{3} x^{3} - 5\,n^{3} x^{4} - 2\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(x^{2} + x^{3} - x^{4} - x^{5} + n x^{2} + 4\,n x^{3} + 3\,n x^{4} - 2\,n x^{5} - 2\,n x^{6} + n^{2} x^{3} + 4\,n^{2} x^{4} + 5\,n^{2} x^{5} + 2\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(1 - 2\,x + x^{2} + 6\,x^{3} + 2\,n + 4\,n x - 7\,n x^{2} - 10\,n x^{3} + 2\,n^{2} x + 6\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(2\,x^{3} + 2\,x^{4} - n x^{2} - 3\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216613
3112302063
41209314875337
51302256407553121963
6142465208042804044142112333
71568615600177802881222841701682967

All 15 rows

Technical details
Node id
n-6a9c2230677418d0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f57b09bec9e86e75

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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: 1,  2;
  2: 1,  6,    6;
  3: 1, 12,   30,    20;
  4: 1, 20,   93,   148,     75;
  5: 1, 30,  225,   640,    755,    312;
  6: 1, 42,  465,  2080,   4280,   4044,    1421;
  7: 1, 56,  861,  5600,  17780,  28812,   22841,    7016;
  8: 1, 72, 1470, 13160,  59955, 147756,  198359,  136056,  37260;
  9: 1, 90, 2358, 27888, 173355, 606186, 1223635, 1407168, 853452, 211470;
%F Formula
The row polynomials satisfy (n+(2*n-2)*x)*P(n,x) = -(n+1+(2*n^2+2*n-2)*x+(n^3+6*n^2-8*n-5)*x^2+(4*n^3-14*n)*x^3+(5*n^3-10*n^2-n+2)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n+1)*x^2+(n^2+4*n+1)*x^3+(4*n^2+3*n-1)*x^4+(5*n^2-2*n-1)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (2*n+1+(2*n^2+4*n-2)*x+(6*n^2-7*n+1)*x^2+(4*n^2-10*n+6)*x^3)*P(n-1,x) - (n*x^2+(3*n-2)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n - 2)*x + (n^3 + 6*n^2 - 8*n - 5)*x^2 + (4*n^3 - 14*n)*x^3 + (5*n^3 - 10*n^2 - n + 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n - 1)*x^4 + (5*n^2 - 2*n - 1)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (2*n + 1 + (2*n^2 + 4*n - 2)*x + (6*n^2 - 7*n + 1)*x^2 + (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] - (n*x^2 + (3*n - 2)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n + (2*n - 2)*x)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 + 2*n - 2)*x + (n^3 + 6*n^2 - 8*n - 5)*x^2 + (4*n^3 - 14*n)*x^3 + (5*n^3 - 10*n^2 - n + 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n + 1)*x^2 + (n^2 + 4*n + 1)*x^3 + (4*n^2 + 3*n - 1)*x^4 + (5*n^2 - 2*n - 1)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (2*n + 1 + (2*n^2 + 4*n - 2)*x + (6*n^2 - 7*n + 1)*x^2 + (4*n^2 - 10*n + 6)*x^3)*v[i - 1] - (n*x^2 + (3*n - 2)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n + (2*n - 2)*x)); 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. [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.