Real-rooted atlas

A063967 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(14 + 69\,x + 126\,x^{2} + 101\,x^{3} + 30\,x^{4} - 6\,n - 30\,n x - 57\,n x^{2} - 48\,n x^{3} - 15\,n x^{4}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 75\,x^{2} + 97\,x^{3} + 61\,x^{4} + 15\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2257
36273063
416108240175539
5443921299189510254655
6120133258741285513950600040131
7328435623970699231139859840035125346087
889613856913923331847248409406506740002056252984443
924484320033213614527443952881684981073802254519125120375025736319
1066881326241164528594156019408410420851256056625055771875298125007046875221936435
111827240222439710882315220088260270229845807414031650509323875409426875194178125412531251913863511
12499201207872132441608685777637851192011507298842490062760383473912541183325002937131250125178750024150000016504154667
1313638435976964337678431604948815497541005387293872136120602392518589038533864639750322668887502067811875080016000001413765625142323171823
14372608106424961399350721121446928611100471223908690332690883841461492856489552411888841002877332618752463458375001433249625005078496875082763281251227320368099
1510179843129811244570198438962941602335798260010157551810833040416803481660589044315408968263652208926651925236790564187518404635875009804240750003203818593754845046875010583766982215
Technical details
Node id
n-4ce86ae0bb1ec232 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6d616c6e46c7ff69

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:    2,      5;
   3:    6,     27,      30;
   4:   16,    108,     240,     175;
   5:   44,    392,    1299,    1895,     1025;
   6:  120,   1332,    5874,   12855,    13950,     6000;
   7:  328,   4356,   23970,   69923,   113985,    98400,    35125;
   8:  896,  13856,   91392,  333184,   724840,   940650,   674000,   205625;
   9: 2448,  43200,  332136, 1452744,  3952881,  6849810,  7380225,  4519125,  1203750;
  10: 6688, 132624, 1164528, 5941560, 19408410, 42085125, 60566250, 55771875, 29812500, 7046875;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(6*n-14+(30*n-69)*x+(57*n-126)*x^2+(48*n-101)*x^3+(15*n-30)*x^4)*P(n-1,x) + (4+28*x+75*x^2+97*x^3+61*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[(-(6*n - 14 + (30*n - 69)*x + (57*n - 126)*x^2 + (48*n - 101)*x^3 + (15*n - 30)*x^4)*P[n - 1, x] + (4 + 28*x + 75*x^2 + 97*x^3 + 61*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] = (-(6*n - 14 + (30*n - 69)*x + (57*n - 126)*x^2 + (48*n - 101)*x^3 + (15*n - 30)*x^4)*v[i - 1] + (4 + 28*x + 75*x^2 + 97*x^3 + 61*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

References

  1. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741