Real-rooted atlas

A081578 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(233 + 1749\,x + 5250\,x^{2} + 8170\,x^{3} + 7005\,x^{4} + 3153\,x^{5} + 584\,x^{6} - 315\,n - 2370\,n x - 7233\,n x^{2} - 11505\,n x^{3} - 10092\,n x^{4} - 4641\,n x^{5} - 876\,n x^{6} + 100\,n^{2} + 720\,n^{2} x + 2172\,n^{2} x^{2} + 3488\,n^{2} x^{3} + 3132\,n^{2} x^{4} + 1488\,n^{2} x^{5} + 292\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(184 + 1939\,x + 8103\,x^{2} + 17870\,x^{3} + 22790\,x^{4} + 16959\,x^{5} + 6859\,x^{6} + 1168\,x^{7} - 80\,n - 848\,n x - 3597\,n x^{2} - 8092\,n x^{3} - 10558\,n x^{4} - 8052\,n x^{5} - 3341\,n x^{6} - 584\,n x^{7}\right)\,P_{n-1}'(x) + \left(16 + 224\,x + 1273\,x^{2} + 3857\,x^{3} + 6910\,x^{4} + 7586\,x^{5} + 5029\,x^{6} + 1853\,x^{7} + 292\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
27353779
320150348245763
4616102166320516697711
518222751091624995272141128576867
654782054958715399525847722213576453769399
71640287002092568219501874488247866017594245176857691803

All 15 rows

Technical details
Node id
n-5364558beebadbdf · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-620515e8c92e1339

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:     2,      5;
  2:     7,     35,      37;
  3:    20,    150,     348,      245;
  4:    61,    610,    2166,     3205,     1669;
  5:   182,   2275,   10916,    24995,    27214,     11285;
  6:   547,   8205,   49587,   153995,   258477,    222135,     76453;
  7:  1640,  28700,  209256,   821950,  1874488,   2478660,   1759424,    517685;
  8:  4921,  98420,  840412,  3994430, 11539126,  20721260,  22575256,  13644905,   3505861;
  9: 14762, 332145, 3250872, 18140010, 63514188, 144554850, 213708840, 197854425, 104114610, 23741525;
%F Formula
The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (100*n^2-315*n+233+(720*n^2-2370*n+1749)*x+(2172*n^2-7233*n+5250)*x^2+(3488*n^2-11505*n+8170)*x^3+(3132*n^2-10092*n+7005)*x^4+(1488*n^2-4641*n+3153)*x^5+(292*n^2-876*n+584)*x^6)*P(n-1,x) - (80*n-184+(848*n-1939)*x+(3597*n-8103)*x^2+(8092*n-17870)*x^3+(10558*n-22790)*x^4+(8052*n-16959)*x^5+(3341*n-6859)*x^6+(584*n-1168)*x^7)*d/dx P(n-1,x) + (16+224*x+1273*x^2+3857*x^3+6910*x^4+7586*x^5+5029*x^6+1853*x^7+292*x^8)*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[((100*n^2 - 315*n + 233 + (720*n^2 - 2370*n + 1749)*x + (2172*n^2 - 7233*n + 5250)*x^2 + (3488*n^2 - 11505*n + 8170)*x^3 + (3132*n^2 - 10092*n + 7005)*x^4 + (1488*n^2 - 4641*n + 3153)*x^5 + (292*n^2 - 876*n + 584)*x^6)*P[n - 1, x] - (80*n - 184 + (848*n - 1939)*x + (3597*n - 8103)*x^2 + (8092*n - 17870)*x^3 + (10558*n - 22790)*x^4 + (8052*n - 16959)*x^5 + (3341*n - 6859)*x^6 + (584*n - 1168)*x^7)*D[P[n - 1, x], x] + (16 + 224*x + 1273*x^2 + 3857*x^3 + 6910*x^4 + 7586*x^5 + 5029*x^6 + 1853*x^7 + 292*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)]];
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] = ((100*n^2 - 315*n + 233 + (720*n^2 - 2370*n + 1749)*x + (2172*n^2 - 7233*n + 5250)*x^2 + (3488*n^2 - 11505*n + 8170)*x^3 + (3132*n^2 - 10092*n + 7005)*x^4 + (1488*n^2 - 4641*n + 3153)*x^5 + (292*n^2 - 876*n + 584)*x^6)*v[i - 1] - (80*n - 184 + (848*n - 1939)*x + (3597*n - 8103)*x^2 + (8092*n - 17870)*x^3 + (10558*n - 22790)*x^4 + (8052*n - 16959)*x^5 + (3341*n - 6859)*x^6 + (584*n - 1168)*x^7)*deriv(v[i - 1], x) + (16 + 224*x + 1273*x^2 + 3857*x^3 + 6910*x^4 + 7586*x^5 + 5029*x^6 + 1853*x^7 + 292*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)); 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

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