Real-rooted atlas

A124932 → 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

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(8 + 60\,x + 186\,x^{2} + 305\,x^{3} + 279\,x^{4} + 135\,x^{5} + 27\,x^{6}\right)\,P_{n-3}(x) + \left(-12 - 72\,x - 171\,x^{2} - 201\,x^{3} - 117\,x^{4} - 27\,x^{5}\right)\,P_{n-2}(x) + \left(6 + 27\,x + 45\,x^{2} + 33\,x^{3} + 9\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 8\,x\)
\(P_{2}(x) = 18 + 57\,x + 45\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
2185745120
356264414216950
41601000234024309456875
54323360104401620012555388846875
61120104164032083160963905953515309306250
7281630464141120362880559440517104265356583201937500
86912852484596481415232272160033475682571912112849221651311953125
9166402304001416960508032011702880179625601837080012072240462550578732072265625
103942460544041817601710720045904320844240321077753609430344054128250184036052814669429687500
1192160155443211911680547430401676505603592512005496543366004627204590075602338340407144929099202322519531250
12212992391372832947200168030720578223360141442329625219434243302544960315242928021391484409795048122717434983454366514599609375
13487424969113688897536498161664190270080052306974721064820556816252524288185993327521576214640096149093043997853496101558375111904278483740234375
141105920236544002348236801434101760601945344018369711360420323904007325645184097725307680993015223207565830272041912980200159591732303738687435406552365476074218750
Technical details
Node id
n-3cb0df9d7da51f9d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8ebc68b94e36a726

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:     5,      8;
  2:    18,     57,      45;
  3:    56,    264,     414,     216;
  4:   160,   1000,    2340,    2430,      945;
  5:   432,   3360,   10440,   16200,    12555,     3888;
  6:  1120,  10416,   40320,   83160,    96390,    59535,    15309;
  7:  2816,  30464,  141120,  362880,   559440,   517104,   265356,    58320;
  8:  6912,  85248,  459648, 1415232,  2721600,  3347568,  2571912,  1128492,  216513;
  9: 16640, 230400, 1416960, 5080320, 11702880, 17962560, 18370800, 12072240, 4625505, 787320;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (8+60*x+186*x^2+305*x^3+279*x^4+135*x^5+27*x^6)*P(n-3,x) - (12+72*x+171*x^2+201*x^3+117*x^4+27*x^5)*P(n-2,x) + (6+27*x+45*x^2+33*x^3+9*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 18 + 57*x + 45*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 8*x;
P[2, x_] := 18 + 57*x + 45*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*P[n - 3, x] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*P[n - 2, x] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*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] = 5 + 8*x; v[3] = 18 + 57*x + 45*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((8 + 60*x + 186*x^2 + 305*x^3 + 279*x^4 + 135*x^5 + 27*x^6)*v[i - 3] - (12 + 72*x + 171*x^2 + 201*x^3 + 117*x^4 + 27*x^5)*v[i - 2] + (6 + 27*x + 45*x^2 + 33*x^3 + 9*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

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.