Real-rooted atlas

A113413 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 4\,x + 6\,x^{2} + 4\,x^{3} + x^{4} + n + 4\,n x + 6\,n x^{2} + 4\,n x^{3} + n x^{4}\right)\,P_{n}(x) = \left(1 + 4\,x + 10\,x^{2} + 36\,x^{3} + 89\,x^{4} + 112\,x^{5} + 68\,x^{6} + 16\,x^{7} + n + 7\,n x + 6\,n x^{2} - 50\,n x^{3} - 155\,n x^{4} - 189\,n x^{5} - 108\,n x^{6} - 24\,n x^{7} + 3\,n^{2} x + 20\,n^{2} x^{2} + 58\,n^{2} x^{3} + 92\,n^{2} x^{4} + 83\,n^{2} x^{5} + 40\,n^{2} x^{6} + 8\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(-x + x^{2} + 50\,x^{3} + 202\,x^{4} + 363\,x^{5} + 341\,x^{6} + 164\,x^{7} + 32\,x^{8} - n x - 14\,n x^{2} - 70\,n x^{3} - 176\,n x^{4} - 249\,n x^{5} - 202\,n x^{6} - 88\,n x^{7} - 16\,n x^{8}\right)\,P_{n-1}'(x) + \left(x^{2} + 13\,x^{3} + 58\,x^{4} + 130\,x^{5} + 165\,x^{6} + 121\,x^{7} + 48\,x^{8} + 8\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
216714
319231750
4112487641178
51158220423399634
61181254287656822392258
71211777751907264919355778042
812423812724010765686805368139328642
912730819467506183582854827312146413363102010
10130387282412908386527693010074483313394068119363314
1113347539332081073920180098301510340661247985104935196011293962
1213657253003188713127237949877728811213551113364723624276996473214608514
131396786952468952197897367721790302315753939982453538998207716472584911424316413466
141427938916666713507661339509377461678949291221196613770207109921485880797189025827580758457426
Technical details
Node id
n-b4fe0b3e9ae272b8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-b42d277a4b716fed

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,  6,   7;
  3: 1,  9,  23,   17;
  4: 1, 12,  48,   76,   41;
  5: 1, 15,  82,  204,  233,    99;
  6: 1, 18, 125,  428,  765,   682,   239;
  7: 1, 21, 177,  775, 1907,  2649,  1935,   577;
  8: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;
  9: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;
%F Formula
The row polynomials satisfy (n+1+(4*n+4)*x+(6*n+6)*x^2+(4*n+4)*x^3+(n+1)*x^4)*P(n,x) = (n+1+(3*n^2+7*n+4)*x+(20*n^2+6*n+10)*x^2+(58*n^2-50*n+36)*x^3+(92*n^2-155*n+89)*x^4+(83*n^2-189*n+112)*x^5+(40*n^2-108*n+68)*x^6+(8*n^2-24*n+16)*x^7)*P(n-1,x) - ((n+1)*x+(14*n-1)*x^2+(70*n-50)*x^3+(176*n-202)*x^4+(249*n-363)*x^5+(202*n-341)*x^6+(88*n-164)*x^7+(16*n-32)*x^8)*d/dx P(n-1,x) + (x^2+13*x^3+58*x^4+130*x^5+165*x^6+121*x^7+48*x^8+8*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 + 1 + (3*n^2 + 7*n + 4)*x + (20*n^2 + 6*n + 10)*x^2 + (58*n^2 - 50*n + 36)*x^3 + (92*n^2 - 155*n + 89)*x^4 + (83*n^2 - 189*n + 112)*x^5 + (40*n^2 - 108*n + 68)*x^6 + (8*n^2 - 24*n + 16)*x^7)*P[n - 1, x] - ((n + 1)*x + (14*n - 1)*x^2 + (70*n - 50)*x^3 + (176*n - 202)*x^4 + (249*n - 363)*x^5 + (202*n - 341)*x^6 + (88*n - 164)*x^7 + (16*n - 32)*x^8)*D[P[n - 1, x], x] + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*D[P[n - 1, x], {x, 2}])/(n + 1 + (4*n + 4)*x + (6*n + 6)*x^2 + (4*n + 4)*x^3 + (n + 1)*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 + 1 + (3*n^2 + 7*n + 4)*x + (20*n^2 + 6*n + 10)*x^2 + (58*n^2 - 50*n + 36)*x^3 + (92*n^2 - 155*n + 89)*x^4 + (83*n^2 - 189*n + 112)*x^5 + (40*n^2 - 108*n + 68)*x^6 + (8*n^2 - 24*n + 16)*x^7)*v[i - 1] - ((n + 1)*x + (14*n - 1)*x^2 + (70*n - 50)*x^3 + (176*n - 202)*x^4 + (249*n - 363)*x^5 + (202*n - 341)*x^6 + (88*n - 164)*x^7 + (16*n - 32)*x^8)*deriv(v[i - 1], x) + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (4*n + 4)*x + (6*n + 6)*x^2 + (4*n + 4)*x^3 + (n + 1)*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