Real-rooted atlas

A357367 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - x + 5\,x^{2} + 9\,x^{3} + 4\,x^{4} + n + 3\,n x - n x^{2} - 7\,n x^{3} - 4\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 18\,x^{3} + 14\,x^{4} + 4\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1022
2061824
3024168264456
4012014405040540012000
507201296074880174240141840404640
605040126000105840039816006930000455112016652160
704032013305601512000081244800225590400313407360172529280809262720
8036288015240960223534080160030080062741952001377782784015921722880754536384045358548480
9036288001886976003454617600314979840001628750592005009195520009087676416008987956992003739442112002880447091200
1003991680025147584005604318720063036610560041141447424001656722833920041838873753600646235428608005585529882240020711190931200204399243417600
11047900160035925120000955608192000129474132480001034068654080005221443161088001717077035520000367443564364800049392944536320003790797107328000126778455175680016028879398963200
12062270208005479778304001712430720000027433140134400026200190016000001609328691210240066030232607539200183206113009920000339906880849536000404354210996736000279050723413862400849917911594752001376545267864166400
13087178291200889218570240032221096427520060126867440640006744853622707200049274111711784960024480898613563392008441225304409497600202353807799503360003312170484150528000035331053639635353600221489286668762112006193091146059417600128486007770209689600
14013076743680001529979010560006357912777216000136508127275520000177292267767014400015137255335378176000890698700703651840003705399046458869760001100339841683706624000232043302749805747200033969486480209280000003283865247386820096000188583891959746483200048736176191602099200012951450419177926656000
Technical details
Node id
n-cb6c3887438e662f · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cb8a225300b7095e

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 0)
  • 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: 1;
  1: 0,       2;
  2: 0,       6,        18;
  3: 0,      24,       168,        264;
  4: 0,     120,      1440,       5040,        5400;
  5: 0,     720,     12960,      74880,      174240,       141840;
  6: 0,    5040,    126000,    1058400,     3981600,      6930000,      4551120;
  7: 0,   40320,   1330560,   15120000,    81244800,    225590400,    313407360,    172529280;
  8: 0,  362880,  15240960,  223534080,  1600300800,   6274195200,  13777827840,  15921722880,   7545363840;
  9: 0, 3628800, 188697600, 3454617600, 31497984000, 162875059200, 500919552000, 908767641600, 898795699200, 373944211200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n-1+(3*n-1)*x-(n-5)*x^2-(7*n-9)*x^3-(4*n-4)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 1)*x - (n - 5)*x^2 - (7*n - 9)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n - 1 + (3*n - 1)*x - (n - 5)*x^2 - (7*n - 9)*x^3 - (4*n - 4)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.