Real-rooted atlas

A133214 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(11 + 56\,x + 94\,x^{2} + 64\,x^{3} + 15\,x^{4} + 8\,n + 43\,n x + 77\,n x^{2} + 57\,n x^{3} + 15\,n x^{4}\right)\,P_{n-1}(x) + \left(-1 - 11\,x - 42\,x^{2} - 70\,x^{3} - 53\,x^{4} - 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 58\,x + 61\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0135861132
1634178855951960
23212820903012420614530736
31683184357899016495016721565527496800
489891179666333151776580273979521963667129098185920
5486397438694800243169177053511028542830151283798417863667136694400
6265729464077634981660148488760387744070636984376642007324362435080876157452305163520
7146256328717695247630860122973438038723069708010304722108742229409331525980458812399598372669539167756800
880974531765746901714211865975008412035955161130897564399481534852160101773804572401324756676255768596029011112210781669500081152
94504671910800998011165800982574727690015315893935830924117914874190781053062627776347113902793027700155184605522524572131834305312614211918711499511326720
1025159596965788638847814936665855739816022026573679480158916616508184215810579855643794139256168848061715233455427557312936602534061319897889791730991481437751935361198313797750784
1114099336193992902169151765242841040662521370902158398624310581720818056449226956244688316467906564225100715784360296005802067200980045639592939163178343550524468242111358010021895164433809947753431589272657920
127923848253241606821750339453350281529118500540220170315603663205718302243757098225180942496009553284632815528409560131903296975129368490932580901299214275477218193860025962266204609240383468175137667419613535018860925921572559551303305019392
1344642381823145812917848522069328383935205268388038805131172557812159560984883317127772130333855437287556917784419114905117229795737699645186067106583252895225648570013831965205260731061470175135523255074850825612922293737958751697309936802082521688330848413951036023797669068800
14252055236609878004568065614242069390860014276527659525609896141235637020502851595870784161936994765141676245768197945562738401341651888588224070244480377092797560034775887606753493203819628613724203280317475930212966070019298085336849004008088197877019086002087699892073533602499426920029286518063104593293803520
Technical details
Node id
n-79e8e9ac32c0a215 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c57d05a45d068312

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,         58,          61;
  1:       63,        417,         885,         595;
  2:      321,       2820,        9030,       12420,         6145;
  3:     1683,      18435,       78990,      164950,       167215,        65527;
  4:     8989,     117966,      633315,     1776580,      2739795,      2196366,        712909;
  5:    48639,     743869,     4800243,    16917705,     35110285,     42830151,      28379841,       7863667;
  6:   265729,    4640776,    34981660,   148488760,    387744070,    636984376,     642007324,     362435080,      87615745;
  7:  1462563,   28717695,   247630860,  1229734380,   3872306970,   8010304722,   10874222940,    9331525980,    4588123995,     983726695;
  8:  8097453,  176574690,  1714211865,  9750084120,  35955161130,  89756439948,  153485216010,  177380457240,  132475667625,   57685960290,  11112210781;
  9: 45046719, 1080099801, 11658009825, 74727690015, 315893935830, 924117914874, 1907810530626, 2777634711390, 2793027700155, 1846055225245, 721318343053, 126142119187;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (8*n+11+(43*n+56)*x+(77*n+94)*x^2+(57*n+64)*x^3+(15*n+15)*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 >= 1, with P(0,x) = 13 + 58*x + 61*x^2.
%t Mathematica
P[0, x_] := 13 + 58*x + 61*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 11 + (43*n + 56)*x + (77*n + 94)*x^2 + (57*n + 64)*x^3 + (15*n + 15)*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 + 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 + 58*x + 61*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 11 + (43*n + 56)*x + (77*n + 94)*x^2 + (57*n + 64)*x^3 + (15*n + 15)*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 + 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