A249250 → x ↦ x/(x+1)

OEIS matches ?

• A108981: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,  10;
  3: 1, 12,  36,   28;
  4: 1, 16,  78,  136,    76;
  5: 1, 20, 136,  388,   476,   208;
  6: 1, 24, 210,  848,  1680,  1584,   568;
  7: 1, 28, 300, 1580,  4424,  6672,  5104,  1552;
  8: 1, 32, 406, 2648,  9700, 20576, 24976, 16064,  4240;
  9: 1, 36, 528, 4116, 18756, 51984, 88320, 89568, 49680, 11584;

The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (4*n^2-3*n-1+(28*n^2-41*n+13)*x+(84*n^2-171*n+87)*x^2+(136*n^2-333*n+197)*x^3+(124*n^2-338*n+214)*x^4+(60*n^2-174*n+114)*x^5+(12*n^2-36*n+24)*x^6)*P(n-1,x) - (n+(18*n-19)*x+(98*n-142)*x^2+(256*n-426)*x^3+(369*n-664)*x^4+(302*n-571)*x^5+(132*n-258)*x^6+(24*n-48)*x^7)*d/dx P(n-1,x) + (x+17*x^2+82*x^3+190*x^4+245*x^5+181*x^6+72*x^7+12*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 3*n - 1 + (28*n^2 - 41*n + 13)*x + (84*n^2 - 171*n + 87)*x^2 + (136*n^2 - 333*n + 197)*x^3 + (124*n^2 - 338*n + 214)*x^4 + (60*n^2 - 174*n + 114)*x^5 + (12*n^2 - 36*n + 24)*x^6)*P[n - 1, x] - (n + (18*n - 19)*x + (98*n - 142)*x^2 + (256*n - 426)*x^3 + (369*n - 664)*x^4 + (302*n - 571)*x^5 + (132*n - 258)*x^6 + (24*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 3*n - 1 + (28*n^2 - 41*n + 13)*x + (84*n^2 - 171*n + 87)*x^2 + (136*n^2 - 333*n + 197)*x^3 + (124*n^2 - 338*n + 214)*x^4 + (60*n^2 - 174*n + 114)*x^5 + (12*n^2 - 36*n + 24)*x^6)*v[i - 1] - (n + (18*n - 19)*x + (98*n - 142)*x^2 + (256*n - 426)*x^3 + (369*n - 664)*x^4 + (302*n - 571)*x^5 + (132*n - 258)*x^6 + (24*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 17*x^2 + 82*x^3 + 190*x^4 + 245*x^5 + 181*x^6 + 72*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b8a3e660dc9132dc
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-b8a3e660dc9132dc
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-b8a3e660dc9132dc

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image mobius, shift 0 e-b8a3e660dc9132dc

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-86238c6a50aeb955 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

Related sequences ?

No candidate equalities or related sequences.

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