Real-rooted atlas

A106246 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 4\,x + 6\,x^{2} + 4\,x^{3} + x^{4}\right)\,P_{n}(x) = \left(-x^{4} - 4\,x^{5} - 6\,x^{6} - 4\,x^{7} - x^{8}\right)\,P_{n-4}(x) + \left(4\,x^{3} + 16\,x^{4} + 24\,x^{5} + 16\,x^{6} + 4\,x^{7}\right)\,P_{n-3}(x) + \left(-6\,x^{2} - 24\,x^{3} - 36\,x^{4} - 24\,x^{5} - 6\,x^{6}\right)\,P_{n-2}(x) + \left(4\,x + 16\,x^{2} + 24\,x^{3} + 16\,x^{4} + 4\,x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)
\(P_{2}(x) = 1 + 10\,x + 12\,x^{2}\)
\(P_{3}(x) = 6\,x + 30\,x^{2} + 28\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
21101223
306302864
400186855141
50004013096266
6000075222154451
700000126350232708
80000001965203331049
900000002887384601486
100000000040510106162031
1100000000055013428042696
120000000000726174010273493
1300000000000936221012884434
140000000000001183275815905531
Technical details
Node id
n-2ceed87b380f70ef · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-02d3db95faa2d653

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: 2,  4;
  2: 1, 10, 12;
  3: 0,  6, 30, 28;
  4: 0,  0, 18, 68,  55;
  5: 0,  0,  0, 40, 130,  96;
  6: 0,  0,  0,  0,  75, 222, 154;
  7: 0,  0,  0,  0,   0, 126, 350, 232;
  8: 0,  0,  0,  0,   0,   0, 196, 520, 333;
  9: 0,  0,  0,  0,   0,   0,   0, 288, 738, 460;
%F Formula
The row polynomials satisfy (1+4*x+6*x^2+4*x^3+x^4)*P(n,x) = -(x^4+4*x^5+6*x^6+4*x^7+x^8)*P(n-4,x) + (4*x^3+16*x^4+24*x^5+16*x^6+4*x^7)*P(n-3,x) - (6*x^2+24*x^3+36*x^4+24*x^5+6*x^6)*P(n-2,x) + (4*x+16*x^2+24*x^3+16*x^4+4*x^5)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 1 + 10*x + 12*x^2, P(3,x) = 6*x + 30*x^2 + 28*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 1 + 10*x + 12*x^2;
P[3, x_] := 6*x + 30*x^2 + 28*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(x^4 + 4*x^5 + 6*x^6 + 4*x^7 + x^8)*P[n - 4, x] + (4*x^3 + 16*x^4 + 24*x^5 + 16*x^6 + 4*x^7)*P[n - 3, x] - (6*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 6*x^6)*P[n - 2, x] + (4*x + 16*x^2 + 24*x^3 + 16*x^4 + 4*x^5)*P[n - 1, x])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)]];
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] = 2 + 4*x; v[3] = 1 + 10*x + 12*x^2; v[4] = 6*x + 30*x^2 + 28*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(x^4 + 4*x^5 + 6*x^6 + 4*x^7 + x^8)*v[i - 4] + (4*x^3 + 16*x^4 + 24*x^5 + 16*x^6 + 4*x^7)*v[i - 3] - (6*x^2 + 24*x^3 + 36*x^4 + 24*x^5 + 6*x^6)*v[i - 2] + (4*x + 16*x^2 + 24*x^3 + 16*x^4 + 4*x^5)*v[i - 1])/(1 + 4*x + 6*x^2 + 4*x^3 + x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.