Real-rooted atlas

A114656 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(3 + 10\,x + 11\,x^{2} + 4\,x^{3} + 3\,n + 10\,n x + 11\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 18\,x^{2} + 14\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
311302061
445186252112595
51971092224020166726217
69036280172802352015840422468047
742793580212366022572022968012355227456770149
8207932033508454601937320264264021461449609601830408939707
9103049115341656080641547145626490464288288001947545674680321244672105843409
105188596540876364141441175079362422405443308497922994001921731225605804697685995521273241431
11264672337106550232754340860212080207445056034110236163873192960299909376015156710404514764806019686415517824973
1213648869210649362146997774061225348801691186112032530891008444669603844319538432029224419840131159705603514570752426008576191202877411
137103937311968155009196640640426232540801327017225602923915484164675361433605466685593604639069670402786552524801124662640642738626560030429184002377843390873
1437269351968057676965710294886429148335673610099472423042508342051072459434075443262837141452806417731713536483437574963226108239872009573021286402136128716802190901248029807864423071
15196880151938735977266352366829724196430573757674970223167842072365927632042787334606016670324979301128007641933491272589065669120491542007756802411155289088080994880512001667823575040158840340480376255282112629
Technical details
Node id
n-a82935040c56619d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9a4b425d2a954ae8

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,       4;
   3:     11,      30,       20;
   4:     45,     186,      252,       112;
   5:    197,    1092,     2240,      2016,       672;
   6:    903,    6280,    17280,     23520,     15840,      4224;
   7:   4279,   35802,   123660,    225720,    229680,    123552,     27456;
   8:  20793,  203350,   845460,   1937320,   2642640,   2146144,    960960,    183040;
   9: 103049, 1153416,  5608064,  15471456,  26490464,  28828800,  19475456,   7468032,  1244672;
  10: 518859, 6540876, 36414144, 117507936, 242240544, 330849792, 299400192, 173122560, 58046976, 8599552;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (3*n+3+(10*n+10)*x+(11*n+11)*x^2+(4*n+4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*x^4)*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[((3*n + 3 + (10*n + 10)*x + (11*n + 11)*x^2 + (4*n + 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*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] = ((3*n + 3 + (10*n + 10)*x + (11*n + 11)*x^2 + (4*n + 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*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