Real-rooted atlas

A124932 → reverse → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(1 + 9\,x + 33\,x^{2} + 63\,x^{3} + 66\,x^{4} + 36\,x^{5} + 8\,x^{6}\right)\,P_{n-3}(x) + \left(-3 - 21\,x - 57\,x^{2} - 75\,x^{3} - 48\,x^{4} - 12\,x^{5}\right)\,P_{n-2}(x) + \left(3 + 15\,x + 27\,x^{2} + 21\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 5\,x\)
\(P_{2}(x) = 6 + 21\,x + 18\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1358
26211845
310549656216
415110300360160945
521195720132012004323888
6283151470364050403696112015309
73647626888400156801747210752281658320
845684453617136403206048056448299526912216513
955945720031920907201713602150401728008064016640787320
106612651089055440184800421344665280718080506880211200394242814669
1178165015840910803484809313921774080240768022809601436160540672921609920232
1291210622308143000617760189446442282246918912823680069555203953664135782421299234543665
1310526393057621621610410403603600922521617681664253693442690688020500480106229763354624487424119042784
14120325540950316680168168064864801873872041184000691891208840832084564480587059202795520081715201105920406552365
Technical details
Node id
n-a5aa0193a07deb50 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b144b587414c3e17

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,   5;
  2:  6,  21,   18;
  3: 10,  54,   96,    56;
  4: 15, 110,  300,   360,   160;
  5: 21, 195,  720,  1320,  1200,    432;
  6: 28, 315, 1470,  3640,  5040,   3696,   1120;
  7: 36, 476, 2688,  8400, 15680,  17472,  10752,   2816;
  8: 45, 684, 4536, 17136, 40320,  60480,  56448,  29952,  6912;
  9: 55, 945, 7200, 31920, 90720, 171360, 215040, 172800, 80640, 16640;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+9*x+33*x^2+63*x^3+66*x^4+36*x^5+8*x^6)*P(n-3,x) - (3+21*x+57*x^2+75*x^3+48*x^4+12*x^5)*P(n-2,x) + (3+15*x+27*x^2+21*x^3+6*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 5*x, P(2,x) = 6 + 21*x + 18*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 5*x;
P[2, x_] := 6 + 21*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*P[n - 3, x] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*P[n - 2, x] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*P[n - 1, x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = 3 + 5*x; v[3] = 6 + 21*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + 9*x + 33*x^2 + 63*x^3 + 66*x^4 + 36*x^5 + 8*x^6)*v[i - 3] - (3 + 21*x + 57*x^2 + 75*x^3 + 48*x^4 + 12*x^5)*v[i - 2] + (3 + 15*x + 27*x^2 + 21*x^3 + 6*x^4)*v[i - 1])/(1 + 3*x + 3*x^2 + x^3)); 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.