Real-rooted atlas

A114687 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 22\,x^{2} + 12\,x^{3} + 2\,x^{4} + 2\,n + 5\,n x + 2\,n x^{2} - 3\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 14\,x^{2} + 18\,x^{3} + 10\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24141129
38488445185
4161444164841971257
5324001680316027509038925
66410566000161202238015546427965445
7128268819712705601383201518728775220793491825
8256665660928277760718144110432010017284953681030493768209
95121612817971210106883298176664070483852166475824279858651885929324405
1010243840051072034598401381632034882176573552006136824041246580158279102646723231153133
112048901121408000112780805385600016540339233932236847073840043668636025974168089625756136488691841801065
124096208896378470435312640198063360723469824179721062430984687363713838480303944520016210941125081294527103937314810069497
13819247923299573761068953606941608962963380992871848806418134907648269263325762839157006420781618000100449020882884263798372693519120029657805
1416384108953625718784314402816233602969611493081600393647654409664704192017261628384022486569705621154828094414002329213661875173724163904652661968801519979470140661
Technical details
Node id
n-bd61f7a659e54e52 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-acc791d1c44ba380

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:   4,    14,     11;
  3:   8,    48,     84,      45;
  4:  16,   144,    416,     484,     197;
  5:  32,   400,   1680,    3160,    2750,     903;
  6:  64,  1056,   6000,   16120,   22380,   15546,    4279;
  7: 128,  2688,  19712,   70560,  138320,  151872,   87752,   20793;
  8: 256,  6656,  60928,  277760,  718144, 1104320, 1001728,  495368,  103049;
  9: 512, 16128, 179712, 1010688, 3298176, 6640704, 8385216, 6475824, 2798586, 518859;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (2*n+4+(5*n+16)*x+(2*n+22)*x^2-(3*n-12)*x^3-(2*n-2)*x^4)*P(n-1,x) + (4*x+14*x^2+18*x^3+10*x^4+2*x^5)*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 + (5*n + 16)*x + (2*n + 22)*x^2 - (3*n - 12)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + (5*n + 16)*x + (2*n + 22)*x^2 - (3*n - 12)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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

Candidate equalities ?

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023