Real-rooted atlas

A116412 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4 + 12\,x + 12\,x^{2} + 4\,x^{3} + 4\,n + 12\,n x + 12\,n x^{2} + 4\,n x^{3}\right)\,P_{n}(x) = \left(172 + 1098\,x + 2886\,x^{2} + 4012\,x^{3} + 3120\,x^{4} + 1290\,x^{5} + 222\,x^{6} - 191\,n - 1296\,n x - 3615\,n x^{2} - 5312\,n x^{3} - 4341\,n x^{4} - 1872\,n x^{5} - 333\,n x^{6} + 51\,n^{2} + 342\,n^{2} x + 969\,n^{2} x^{2} + 1476\,n^{2} x^{3} + 1269\,n^{2} x^{4} + 582\,n^{2} x^{5} + 111\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(134 + 1192\,x + 4402\,x^{2} + 8812\,x^{3} + 10378\,x^{4} + 7216\,x^{5} + 2750\,x^{6} + 444\,x^{7} - 52\,n - 470\,n x - 1784\,n x^{2} - 3698\,n x^{3} - 4532\,n x^{4} - 3290\,n x^{5} - 1312\,n x^{6} - 222\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 150\,x + 730\,x^{2} + 1966\,x^{3} + 3220\,x^{4} + 3298\,x^{5} + 2070\,x^{6} + 730\,x^{7} + 111\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
213423388
343213345183784
41429482340253210146976
54693939130862149817469561962080
61549156786551114466017805911585431137552448
7511660594305172847194140019613778467475641725424916224

All 15 rows

Technical details
Node id
n-93a4bd53e722afe2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-02fbd202e4877ba3

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:     4,      6;
  2:    13,     42,      33;
  3:    43,    213,     345,      183;
  4:   142,    948,    2340,     2532,     1014;
  5:   469,   3939,   13086,    21498,    17469,     5619;
  6:  1549,  15678,   65511,   144660,   178059,   115854,     31137;
  7:  5116,  60594,  305172,   847194,  1400196,  1377846,    747564,   172542;
  8: 16897, 229236, 1351356,  4521180,  9389790, 12396588,  10160652,  4727556,   956121;
  9: 55807, 853251, 5763144, 22569984, 56479758, 93658926, 102924432, 72282264, 29439003, 5298231;
%F Formula
The row polynomials satisfy (4*n+4+(12*n+12)*x+(12*n+12)*x^2+(4*n+4)*x^3)*P(n,x) = (51*n^2-191*n+172+(342*n^2-1296*n+1098)*x+(969*n^2-3615*n+2886)*x^2+(1476*n^2-5312*n+4012)*x^3+(1269*n^2-4341*n+3120)*x^4+(582*n^2-1872*n+1290)*x^5+(111*n^2-333*n+222)*x^6)*P(n-1,x) - (52*n-134+(470*n-1192)*x+(1784*n-4402)*x^2+(3698*n-8812)*x^3+(4532*n-10378)*x^4+(3290*n-7216)*x^5+(1312*n-2750)*x^6+(222*n-444)*x^7)*d/dx P(n-1,x) + (13+150*x+730*x^2+1966*x^3+3220*x^4+3298*x^5+2070*x^6+730*x^7+111*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[((51*n^2 - 191*n + 172 + (342*n^2 - 1296*n + 1098)*x + (969*n^2 - 3615*n + 2886)*x^2 + (1476*n^2 - 5312*n + 4012)*x^3 + (1269*n^2 - 4341*n + 3120)*x^4 + (582*n^2 - 1872*n + 1290)*x^5 + (111*n^2 - 333*n + 222)*x^6)*P[n - 1, x] - (52*n - 134 + (470*n - 1192)*x + (1784*n - 4402)*x^2 + (3698*n - 8812)*x^3 + (4532*n - 10378)*x^4 + (3290*n - 7216)*x^5 + (1312*n - 2750)*x^6 + (222*n - 444)*x^7)*D[P[n - 1, x], x] + (13 + 150*x + 730*x^2 + 1966*x^3 + 3220*x^4 + 3298*x^5 + 2070*x^6 + 730*x^7 + 111*x^8)*D[P[n - 1, x], {x, 2}])/(4*n + 4 + (12*n + 12)*x + (12*n + 12)*x^2 + (4*n + 4)*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] = ((51*n^2 - 191*n + 172 + (342*n^2 - 1296*n + 1098)*x + (969*n^2 - 3615*n + 2886)*x^2 + (1476*n^2 - 5312*n + 4012)*x^3 + (1269*n^2 - 4341*n + 3120)*x^4 + (582*n^2 - 1872*n + 1290)*x^5 + (111*n^2 - 333*n + 222)*x^6)*v[i - 1] - (52*n - 134 + (470*n - 1192)*x + (1784*n - 4402)*x^2 + (3698*n - 8812)*x^3 + (4532*n - 10378)*x^4 + (3290*n - 7216)*x^5 + (1312*n - 2750)*x^6 + (222*n - 444)*x^7)*deriv(v[i - 1], x) + (13 + 150*x + 730*x^2 + 1966*x^3 + 3220*x^4 + 3298*x^5 + 2070*x^6 + 730*x^7 + 111*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n + 4 + (12*n + 12)*x + (12*n + 12)*x^2 + (4*n + 4)*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