Real-rooted atlas

A104684 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-15 - 71\,x - 121\,x^{2} - 89\,x^{3} - 24\,x^{4} + 9\,n + 41\,n x + 67\,n x^{2} + 47\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-2 - 16\,x - 48\,x^{2} - 68\,x^{3} - 46\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2358
313423388
4633034772451088
5321205248545028192114176
616831342542390661705101515525190208
78989859503394957086608239955057101279052600704
848639542171257121967224151046020596827854935105106792536030464
926572933833041872687658839480114746310142180920109280220476197209004545504047104
101462563209403811325061724862626441140214698177123303018223644601197203940455591115764995257104278528
118097453128774610916942365384902028010546026330197038752122541861693022351888440128198864254329818610653808673100726755328
1245046719787806855623414434529460872265923675751902017080048063130148013303451271699702649272892751348198027554092979350956149957651435037302784
132515959694799007756417805508342195100567007750453396951937190842136351419372396446613578120564486410086895305521893270013971920324343852131288764841645452920526579564544
1414099336192912907936927669387774615998731620306281867405385176859419189313672757942167656954845780404659474308285655630587261513534452418342914143050096108143612167523471418895174885294599134674944
157923848253176270348646181412309414711447814429324494806027445731549495500773223625050273834996435864028273688712554408067518949030104834826864684382485933813094921741561449724543646473376619323589436347231021134240277467168768
Technical details
Node id
n-9e97c676f5c9e5f6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b3b060ce99fab65a

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       3,        5;
   3:      13,       42,        33;
   4:      63,      303,       477,       245;
   5:     321,     2052,      4854,      5028,       1921;
   6:    1683,    13425,     42390,     66170,      51015,      15525;
   7:    8989,    85950,    339495,    708660,     823995,     505710,     127905;
   8:   48639,   542171,   2571219,   6722415,   10460205,    9682785,    4935105,    1067925;
   9:  265729,  3383304,  18726876,  58839480,  114746310,  142180920,  109280220,   47619720,   9004545;
  10: 1462563, 20940381, 132506172, 486262644, 1140214698, 1771233030, 1822364460, 1197203940, 455591115, 76499525;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (9*n-15+(41*n-71)*x+(67*n-121)*x^2+(47*n-89)*x^3+(12*n-24)*x^4)*P(n-1,x) - (2+16*x+48*x^2+68*x^3+46*x^4+12*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 15 + (41*n - 71)*x + (67*n - 121)*x^2 + (47*n - 89)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((9*n - 15 + (41*n - 71)*x + (67*n - 121)*x^2 + (47*n - 89)*x^3 + (12*n - 24)*x^4)*v[i - 1] - (2 + 16*x + 48*x^2 + 68*x^3 + 46*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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