Real-rooted atlas

A121574 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 4\,n x + 6\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n}(x) = \left(18\,x^{2} + 98\,x^{3} + 212\,x^{4} + 228\,x^{5} + 122\,x^{6} + 26\,x^{7} + n + 3\,n x - 26\,n x^{2} - 153\,n x^{3} - 327\,n x^{4} - 347\,n x^{5} - 184\,n x^{6} - 39\,n x^{7} + 4\,n^{2} x + 26\,n^{2} x^{2} + 77\,n^{2} x^{3} + 128\,n^{2} x^{4} + 122\,n^{2} x^{5} + 62\,n^{2} x^{6} + 13\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(16\,x^{2} + 128\,x^{3} + 404\,x^{4} + 656\,x^{5} + 584\,x^{6} + 272\,x^{7} + 52\,x^{8} - n x - 16\,n x^{2} - 87\,n x^{3} - 234\,n x^{4} - 351\,n x^{5} - 300\,n x^{6} - 137\,n x^{7} - 26\,n x^{8}\right)\,P_{n-1}'(x) + \left(x^{2} + 15\,x^{3} + 73\,x^{4} + 175\,x^{5} + 235\,x^{6} + 181\,x^{7} + 75\,x^{8} + 13\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2171018
3111353380
411576155109356
51191334296423601584
61232069192160255011897048
712729516895479101019841392731360
813140028031167129617448543718912970139536
91355214325220647179814928519169313830542837620864
101396586319382421524644059677140197954275079111414812762528
11143811884962045293579957221215300332784623224190184635646728012291840
121479801197995569524275202866455830971085745714569022128222646655740154332154692416
1315111651577314116688187639571131291988930716972525728586303198250193078238241775097243243353344
1415513662029520144414129227222585273441677708840016116432824617703526668031019388141584768835168350501082798208
Technical details
Node id
n-b48b62a5f27e6dcd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-98ef740d5236d6f3

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: 1,  3;
  2: 1,  7,  10;
  3: 1, 11,  35,   33;
  4: 1, 15,  76,  155,   109;
  5: 1, 19, 133,  429,   642,   360;
  6: 1, 23, 206,  919,  2160,  2550,   1189;
  7: 1, 27, 295, 1689,  5479, 10101,   9841,   3927;
  8: 1, 31, 400, 2803, 11671, 29617,  44854,  37189,  12970;
  9: 1, 35, 521, 4325, 22064, 71798, 149285, 191693, 138305, 42837;
%F Formula
The row polynomials satisfy (n+4*n*x+6*n*x^2+4*n*x^3+n*x^4)*P(n,x) = (n+(4*n^2+3*n)*x+(26*n^2-26*n+18)*x^2+(77*n^2-153*n+98)*x^3+(128*n^2-327*n+212)*x^4+(122*n^2-347*n+228)*x^5+(62*n^2-184*n+122)*x^6+(13*n^2-39*n+26)*x^7)*P(n-1,x) - (n*x+(16*n-16)*x^2+(87*n-128)*x^3+(234*n-404)*x^4+(351*n-656)*x^5+(300*n-584)*x^6+(137*n-272)*x^7+(26*n-52)*x^8)*d/dx P(n-1,x) + (x^2+15*x^3+73*x^4+175*x^5+235*x^6+181*x^7+75*x^8+13*x^9)*d^2/dx^2 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 + (4*n^2 + 3*n)*x + (26*n^2 - 26*n + 18)*x^2 + (77*n^2 - 153*n + 98)*x^3 + (128*n^2 - 327*n + 212)*x^4 + (122*n^2 - 347*n + 228)*x^5 + (62*n^2 - 184*n + 122)*x^6 + (13*n^2 - 39*n + 26)*x^7)*P[n - 1, x] - (n*x + (16*n - 16)*x^2 + (87*n - 128)*x^3 + (234*n - 404)*x^4 + (351*n - 656)*x^5 + (300*n - 584)*x^6 + (137*n - 272)*x^7 + (26*n - 52)*x^8)*D[P[n - 1, x], x] + (x^2 + 15*x^3 + 73*x^4 + 175*x^5 + 235*x^6 + 181*x^7 + 75*x^8 + 13*x^9)*D[P[n - 1, x], {x, 2}])/(n + 4*n*x + 6*n*x^2 + 4*n*x^3 + n*x^4)]];
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 + (4*n^2 + 3*n)*x + (26*n^2 - 26*n + 18)*x^2 + (77*n^2 - 153*n + 98)*x^3 + (128*n^2 - 327*n + 212)*x^4 + (122*n^2 - 347*n + 228)*x^5 + (62*n^2 - 184*n + 122)*x^6 + (13*n^2 - 39*n + 26)*x^7)*v[i - 1] - (n*x + (16*n - 16)*x^2 + (87*n - 128)*x^3 + (234*n - 404)*x^4 + (351*n - 656)*x^5 + (300*n - 584)*x^6 + (137*n - 272)*x^7 + (26*n - 52)*x^8)*deriv(v[i - 1], x) + (x^2 + 15*x^3 + 73*x^4 + 175*x^5 + 235*x^6 + 181*x^7 + 75*x^8 + 13*x^9)*deriv(deriv(v[i - 1], x), x))/(n + 4*n*x + 6*n*x^2 + 4*n*x^3 + n*x^4)); 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

Candidate equalities ?

Same fingerprint ?

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