Real-rooted atlas

A134082 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 19\,x + 16\,x^{2} + 4\,x^{3} - 4\,n - 12\,n x - 12\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 18\,x^{2} + 14\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
25161233
37366032135
496416819280513
5111003606405601921863
6131446601600216015364486561
71519610923360616067204032102422599
8172561680627214560215041971210240230476545
91932424481075230240564486988855296253445120255879
1021400342017280571201290242016002150401497606144011264846369
1123484462026400100320266112502656675840633600394240146432245762775303
122557660723872016632050688011235841824768215424018022401013760344064532489034497
132767678005491226312090604823063044392960625996865894404978688255590479872011468829229255
1429784982875712400400153753644204169664512161441282050048019475456134184966336512183500824576094065057
Technical details
Node id
n-b877b6e17eb7eb9a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-238832ed0536debf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,   4;
  2:  5,  16,   12;
  3:  7,  36,   60,    32;
  4:  9,  64,  168,   192,    80;
  5: 11, 100,  360,   640,   560,   192;
  6: 13, 144,  660,  1600,  2160,  1536,   448;
  7: 15, 196, 1092,  3360,  6160,  6720,  4032,  1024;
  8: 17, 256, 1680,  6272, 14560, 21504, 19712, 10240,  2304;
  9: 19, 324, 2448, 10752, 30240, 56448, 69888, 55296, 25344, 5120;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(4*n-7+(12*n-19)*x+(12*n-16)*x^2+(4*n-4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*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[(-(4*n - 7 + (12*n - 19)*x + (12*n - 16)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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] = (-(4*n - 7 + (12*n - 19)*x + (12*n - 16)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*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

Candidate equalities ?

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.