Real-rooted atlas

A048993 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4\,x + 11\,x^{2} + 10\,x^{3} + 3\,x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-2}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-2}'(x) + \left(-1 + 4\,x^{2} + 4\,x^{3} + x^{4} - n x - 3\,n x^{2} - 3\,n x^{3} - n x^{4}\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} + 6\,x^{3} + 4\,x^{4} + x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1022
20257
302131530
402277252153
50253247393203898
602103752198322028775919
702201217984961523412850414043102
802395619233610866501161897848821147342673
902781175111276274443718427338956195023271159752946946
10021551496564748572148120540717680404127042695336664867857027209687
1102308914171517515061003195832213861628732857647470456779912235972974213597268080926
12026163407624644911645933764183173774450794792714109267732154948470600342172691956276444372803966185
130212309118176723803765208135371101114826630530009666049415214803153424070980934164014997716131727640019089932230999498754
1402245993450912882957959390183505479135535198992313364797405500279202799946902458853176994989256835279173686104551353501382958545360899056943
Technical details
Node id
n-1f0600945ceb000b · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f58116f4a2b7bfc5

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: 0, 2;
  2: 0, 2,   5;
  3: 0, 2,  13,    15;
  4: 0, 2,  27,    72,     52;
  5: 0, 2,  53,   247,    393,    203;
  6: 0, 2, 103,   752,   1983,   2202,    877;
  7: 0, 2, 201,  2179,   8496,  15234,  12850,   4140;
  8: 0, 2, 395,  6192,  33610,  86650, 116189,  78488,  21147;
  9: 0, 2, 781, 17511, 127627, 444371, 842733, 895619, 502327, 115975;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -((n-4)*x+(3*n-11)*x^2+(3*n-10)*x^3+(n-3)*x^4)*P(n-2,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-2,x) - (1+n*x+(3*n-4)*x^2+(3*n-4)*x^3+(n-1)*x^4)*P(n-1,x) + (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n - 4)*x + (3*n - 11)*x^2 + (3*n - 10)*x^3 + (n - 3)*x^4)*P[n - 2, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 2, x], x] - (1 + n*x + (3*n - 4)*x^2 + (3*n - 4)*x^3 + (n - 1)*x^4)*P[n - 1, x] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n - 4)*x + (3*n - 11)*x^2 + (3*n - 10)*x^3 + (n - 3)*x^4)*v[i - 2] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 2], x) - (1 + n*x + (3*n - 4)*x^2 + (3*n - 4)*x^3 + (n - 1)*x^4)*v[i - 1] + (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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. [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.