Real-rooted atlas

A049020 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 12\,x + 11\,x^{2} + 3\,x^{3} - 2\,n - 7\,n x - 8\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 9\,x + 15\,x^{2} + 11\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
26171235
3229112557295
49450910309233092865
545430318069107067079186631205
6243019257634061110241090315693612351374435
714214130243510145110719114379741117551481214885634887095

All 15 rows

Technical details
Node id
n-7dd420d685af7aaf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a251fb5b77cdd8f2

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,       3;
  2:      6,      17,       12;
  3:     22,      91,      125,        57;
  4:     94,     509,     1030,       923,       309;
  5:    454,    3031,     8069,     10706,      7079,      1866;
  6:   2430,   19257,    63406,    111024,    109031,     56936,     12351;
  7:  14214,  130243,   510145,   1107191,   1437974,   1117551,    481214,     88563;
  8:  89918,  934517,  4239110,  10961817,  17673391,  18191949,  11674944,   4270949,   681870;
  9: 610182, 7087855, 36511877, 109473680, 210539363, 269336047, 229186873, 125089978, 39736907, 5597643;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-4+(7*n-12)*x+(8*n-11)*x^2+(3*n-3)*x^3)*P(n-1,x) + (2+9*x+15*x^2+11*x^3+3*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 + (7*n - 12)*x + (8*n - 11)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n - 4 + (7*n - 12)*x + (8*n - 11)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + 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.