Real-rooted atlas

A081580 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(121 + 801\,x + 2190\,x^{2} + 3170\,x^{3} + 2565\,x^{4} + 1101\,x^{5} + 196\,x^{6} - 173\,n - 1152\,n x - 3173\,n x^{2} - 4631\,n x^{3} - 3780\,n x^{4} - 1637\,n x^{5} - 294\,n x^{6} + 54\,n^{2} + 360\,n^{2} x + 998\,n^{2} x^{2} + 1472\,n^{2} x^{3} + 1218\,n^{2} x^{4} + 536\,n^{2} x^{5} + 98\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(156 + 1287\,x + 4487\,x^{2} + 8590\,x^{3} + 9770\,x^{4} + 6611\,x^{5} + 2467\,x^{6} + 392\,x^{7} - 72\,n - 600\,n x - 2115\,n x^{2} - 4096\,n x^{3} - 4714\,n x^{4} - 3228\,n x^{5} - 1219\,n x^{6} - 196\,n x^{7}\right)\,P_{n-1}'(x) + \left(24 + 240\,x + 1033\,x^{2} + 2503\,x^{3} + 3740\,x^{4} + 3534\,x^{5} + 2065\,x^{6} + 683\,x^{7} + 98\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
29271955
32812618487425
4101606133612814513775
534225657580110257890222331625
61189107013963577295836854769111179271375
7408842924191168467970679880586152277676557672305625

All 15 rows

Technical details
Node id
n-dcc08201a93667c6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2f2d1d2ced8d0643

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,      3;
  2:     9,     27,      19;
  3:    28,    126,     184,       87;
  4:   101,    606,    1336,     1281,      451;
  5:   342,   2565,    7580,    11025,     7890,     2223;
  6:  1189,  10701,   39635,    77295,    83685,    47691,    11179;
  7:  4088,  42924,  191168,   467970,   679880,   586152,   277676,    55767;
  8: 14121, 169452,  881688,  2597454,  4737870,  5478624,  3921918,  1589157,  279091;
  9: 48682, 657207, 3912312, 13476582, 29598732, 42979518, 41259168, 25249563, 8938918, 1394943;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (54*n^2-173*n+121+(360*n^2-1152*n+801)*x+(998*n^2-3173*n+2190)*x^2+(1472*n^2-4631*n+3170)*x^3+(1218*n^2-3780*n+2565)*x^4+(536*n^2-1637*n+1101)*x^5+(98*n^2-294*n+196)*x^6)*P(n-1,x) - (72*n-156+(600*n-1287)*x+(2115*n-4487)*x^2+(4096*n-8590)*x^3+(4714*n-9770)*x^4+(3228*n-6611)*x^5+(1219*n-2467)*x^6+(196*n-392)*x^7)*d/dx P(n-1,x) + (24+240*x+1033*x^2+2503*x^3+3740*x^4+3534*x^5+2065*x^6+683*x^7+98*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[((54*n^2 - 173*n + 121 + (360*n^2 - 1152*n + 801)*x + (998*n^2 - 3173*n + 2190)*x^2 + (1472*n^2 - 4631*n + 3170)*x^3 + (1218*n^2 - 3780*n + 2565)*x^4 + (536*n^2 - 1637*n + 1101)*x^5 + (98*n^2 - 294*n + 196)*x^6)*P[n - 1, x] - (72*n - 156 + (600*n - 1287)*x + (2115*n - 4487)*x^2 + (4096*n - 8590)*x^3 + (4714*n - 9770)*x^4 + (3228*n - 6611)*x^5 + (1219*n - 2467)*x^6 + (196*n - 392)*x^7)*D[P[n - 1, x], x] + (24 + 240*x + 1033*x^2 + 2503*x^3 + 3740*x^4 + 3534*x^5 + 2065*x^6 + 683*x^7 + 98*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + 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] = ((54*n^2 - 173*n + 121 + (360*n^2 - 1152*n + 801)*x + (998*n^2 - 3173*n + 2190)*x^2 + (1472*n^2 - 4631*n + 3170)*x^3 + (1218*n^2 - 3780*n + 2565)*x^4 + (536*n^2 - 1637*n + 1101)*x^5 + (98*n^2 - 294*n + 196)*x^6)*v[i - 1] - (72*n - 156 + (600*n - 1287)*x + (2115*n - 4487)*x^2 + (4096*n - 8590)*x^3 + (4714*n - 9770)*x^4 + (3228*n - 6611)*x^5 + (1219*n - 2467)*x^6 + (196*n - 392)*x^7)*deriv(v[i - 1], x) + (24 + 240*x + 1033*x^2 + 2503*x^3 + 3740*x^4 + 3534*x^5 + 2065*x^6 + 683*x^7 + 98*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + 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