Real-rooted atlas

A133214 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(12 + 52\,x + 80\,x^{2} + 52\,x^{3} + 12\,x^{4} + 9\,n + 41\,n x + 67\,n x^{2} + 47\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-2 - 16\,x - 48\,x^{2} - 68\,x^{3} - 46\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 42\,x + 33\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
013423388
1633034772451088
2321205248545028192114176
316831342542390661705101515525190208
48989859503394957086608239955057101279052600704
548639542171257121967224151046020596827854935105106792536030464
626572933833041872687658839480114746310142180920109280220476197209004545504047104
71462563209403811325061724862626441140214698177123303018223644601197203940455591115764995257104278528
88097453128774610916942365384902028010546026330197038752122541861693022351888440128198864254329818610653808673100726755328
945046719787806855623414434529460872265923675751902017080048063130148013303451271699702649272892751348198027554092979350956149957651435037302784
102515959694799007756417805508342195100567007750453396951937190842136351419372396446613578120564486410086895305521893270013971920324343852131288764841645452920526579564544
1114099336192912907936927669387774615998731620306281867405385176859419189313672757942167656954845780404659474308285655630587261513534452418342914143050096108143612167523471418895174885294599134674944
127923848253176270348646181412309414711447814429324494806027445731549495500773223625050273834996435864028273688712554408067518949030104834826864684382485933813094921741561449724543646473376619323589436347231021134240277467168768
1344642381823106388245291511792646743535806464620001953805031075431951311895612490463341428677449359568296221881637751058560704505804512712294923633305117308191471240778168202409355625415395226874218514564631252676053147999084293853161693718472561183611081064448
1425205523660964064647332007609200187788056057477620536028668482283768601079133257404238430925598354731080688232422361484001201951942370279101652682387918962401783071319887217361493503802271280809520145338552758044642679816878640145227135202138802928049987418640275621102802945884741809748967424
Technical details
Node id
n-c5f47b56c99801fb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b3b060ce99fab65a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:       13,        42,         33;
  1:       63,       303,        477,         245;
  2:      321,      2052,       4854,        5028,        1921;
  3:     1683,     13425,      42390,       66170,       51015,        15525;
  4:     8989,     85950,     339495,      708660,      823995,       505710,       127905;
  5:    48639,    542171,    2571219,     6722415,    10460205,      9682785,      4935105,      1067925;
  6:   265729,   3383304,   18726876,    58839480,   114746310,    142180920,    109280220,     47619720,      9004545;
  7:  1462563,  20940381,  132506172,   486262644,  1140214698,   1771233030,   1822364460,   1197203940,    455591115,     76499525;
  8:  8097453, 128774610,  916942365,  3849020280, 10546026330,  19703875212,  25418616930,  22351888440,  12819886425,   4329818610,   653808673;
  9: 45046719, 787806855, 6234144345, 29460872265, 92367575190, 201708004806, 313014801330, 345127169970, 264927289275, 134819802755, 40929793509, 5614995765;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (9*n+12+(41*n+52)*x+(67*n+80)*x^2+(47*n+52)*x^3+(12*n+12)*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 >= 1, with P(0,x) = 13 + 42*x + 33*x^2.
%t Mathematica
P[0, x_] := 13 + 42*x + 33*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 12 + (41*n + 52)*x + (67*n + 80)*x^2 + (47*n + 52)*x^3 + (12*n + 12)*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 + 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] = 13 + 42*x + 33*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9*n + 12 + (41*n + 52)*x + (67*n + 80)*x^2 + (47*n + 52)*x^3 + (12*n + 12)*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 + 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

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