A062196 → x ↦ x/(x+1)

OEIS matches ?

• A035029: 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,  10,   15;
  3: 1,  18,   63,    56;
  4: 1,  28,  168,   336,    210;
  5: 1,  40,  360,  1200,   1650,    792;
  6: 1,  54,  675,  3300,   7425,   7722,    3003;
  7: 1,  70, 1155,  7700,  25025,  42042,   35035,   11440;
  8: 1,  88, 1848, 16016,  70070, 168168,  224224,  155584,  43758;
  9: 1, 108, 2808, 30576, 171990, 550368, 1039584, 1145664, 680238, 167960;

The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^2+2*n+(4*n^2+10*n+4)*x+(5*n^2+14*n+8)*x^2+(2*n^2+6*n+4)*x^3)*P(n-1,x) + ((2*n+2)*x+(6*n+6)*x^2+(6*n+6)*x^3+(2*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (4*n^2 + 10*n + 4)*x + (5*n^2 + 14*n + 8)*x^2 + (2*n^2 + 6*n + 4)*x^3)*P[n - 1, x] + ((2*n + 2)*x + (6*n + 6)*x^2 + (6*n + 6)*x^3 + (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (4*n^2 + 10*n + 4)*x + (5*n^2 + 14*n + 8)*x^2 + (2*n^2 + 6*n + 4)*x^3)*v[i - 1] + ((2*n + 2)*x + (6*n + 6)*x^2 + (6*n + 6)*x^3 + (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^2)); 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-f6a7bd41834fc468
• 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-f6a7bd41834fc468
• 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-f6a7bd41834fc468

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-f6a7bd41834fc468

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-32e7e43d5a9debda root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

• A033282 → θ+1 (15 rows compared; no all-index equality certificate was found)

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