Real-rooted atlas

A104684 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-13 - 73\,x - 137\,x^{2} - 107\,x^{3} - 30\,x^{4} + 8\,n + 43\,n x + 77\,n x^{2} + 57\,n x^{3} + 15\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 11\,x - 42\,x^{2} - 70\,x^{3} - 53\,x^{4} - 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
23710
3135861132
4634178855951960
53212820903012420614530736
61683184357899016495016721565527496800
789891179666333151776580273979521963667129098185920
8486397438694800243169177053511028542830151283798417863667136694400
9265729464077634981660148488760387744070636984376642007324362435080876157452305163520
10146256328717695247630860122973438038723069708010304722108742229409331525980458812399598372669539167756800
1180974531765746901714211865975008412035955161130897564399481534852160101773804572401324756676255768596029011112210781669500081152
124504671910800998011165800982574727690015315893935830924117914874190781053062627776347113902793027700155184605522524572131834305312614211918711499511326720
1325159596965788638847814936665855739816022026573679480158916616508184215810579855643794139256168848061715233455427557312936602534061319897889791730991481437751935361198313797750784
1414099336193992902169151765242841040662521370902158398624310581720818056449226956244688316467906564225100715784360296005802067200980045639592939163178343550524468242111358010021895164433809947753431589272657920
157923848253241606821750339453350281529118500540220170315603663205718302243757098225180942496009553284632815528409560131903296975129368490932580901299214275477218193860025962266204609240383468175137667419613535018860925921572559551303305019392
Technical details
Node id
n-11a41503366af691 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c57d05a45d068312

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,        7;
   3:      13,       58,        61;
   4:      63,      417,       885,        595;
   5:     321,     2820,      9030,      12420,       6145;
   6:    1683,    18435,     78990,     164950,     167215,      65527;
   7:    8989,   117966,    633315,    1776580,    2739795,    2196366,      712909;
   8:   48639,   743869,   4800243,   16917705,   35110285,   42830151,    28379841,    7863667;
   9:  265729,  4640776,  34981660,  148488760,  387744070,  636984376,   642007324,  362435080,   87615745;
  10: 1462563, 28717695, 247630860, 1229734380, 3872306970, 8010304722, 10874222940, 9331525980, 4588123995, 983726695;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (8*n-13+(43*n-73)*x+(77*n-137)*x^2+(57*n-107)*x^3+(15*n-30)*x^4)*P(n-1,x) - (1+11*x+42*x^2+70*x^3+53*x^4+15*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[((8*n - 13 + (43*n - 73)*x + (77*n - 137)*x^2 + (57*n - 107)*x^3 + (15*n - 30)*x^4)*P[n - 1, x] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*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] = ((8*n - 13 + (43*n - 73)*x + (77*n - 137)*x^2 + (57*n - 107)*x^3 + (15*n - 30)*x^4)*v[i - 1] - (1 + 11*x + 42*x^2 + 70*x^3 + 53*x^4 + 15*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