basis-map coefficients of A026729 → basis A056242 → x ↦ x/(x+1)

OEIS matches ?

• A086405: 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:
   1: 1;
   2: 1,  3;
   3: 1,  7,   10;
   4: 1, 12,   37,   34;
   5: 1, 18,   89,  172,   116;
   6: 1, 25,  175,  531,   744,    396;
   7: 1, 33,  305, 1287,  2802,   3076,   1352;
   8: 1, 42,  490, 2688,  8065,  13670,  12332,   4616;
   9: 1, 52,  742, 5068, 19609,  45544,  63160,  48352,  15760;
  10: 1, 63, 1074, 8862, 42381, 126283, 239024, 280392, 186416, 53808;

The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n-1+(n+3)*x-(5*n-17)*x^2-(9*n-21)*x^3-(4*n-8)*x^4)*P(n-1,x) + (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (n + 3)*x - (5*n - 17)*x^2 - (9*n - 21)*x^3 - (4*n - 8)*x^4)*P[n - 1, x] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n - 1 + (n + 3)*x - (5*n - 17)*x^2 - (9*n - 21)*x^3 - (4*n - 8)*x^4)*v[i - 1] + (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-fc6a306274ee3581
• Coefficients of one sign. Every nonzero row, n >= 1: 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-fc6a306274ee3581
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-fc6a306274ee3581

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-fc6a306274ee3581

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A056242 e-3cb5f3d64c4bb317

      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
        no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.