A152842 → x ↦ x/(x+1)

OEIS matches ?

• A101553: 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,  6,   8;
  3: 1, 10,  32,   32;
  4: 1, 12,  52,   96,    64;
  5: 1, 16, 100,  304,   448,   256;
  6: 1, 18, 132,  504,  1056,  1152,   512;
  7: 1, 22, 204, 1032,  3072,  5376,  5120,  2048;
  8: 1, 24, 248, 1440,  5136, 11520, 15872, 12288,  4096;
  9: 1, 28, 344, 2432, 10896, 32064, 61952, 75776, 53248, 16384;

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+(6*n+1)*x+(34*n-19)*x^2+(66*n-53)*x^3+(54*n-50)*x^4+(16*n-16)*x^5)*P(n-1,x) - (2*x+18*x^2+58*x^3+86*x^4+60*x^5+16*x^6)*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[((1 + (6*n + 1)*x + (34*n - 19)*x^2 + (66*n - 53)*x^3 + (54*n - 50)*x^4 + (16*n - 16)*x^5)*P[n - 1, x] - (2*x + 18*x^2 + 58*x^3 + 86*x^4 + 60*x^5 + 16*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
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] = ((1 + (6*n + 1)*x + (34*n - 19)*x^2 + (66*n - 53)*x^3 + (54*n - 50)*x^4 + (16*n - 16)*x^5)*v[i - 1] - (2*x + 18*x^2 + 58*x^3 + 86*x^4 + 60*x^5 + 16*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + 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-ff2584c8c316f8a8
• 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-ff2584c8c316f8a8
• 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-ff2584c8c316f8a8

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-ff2584c8c316f8a8

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-596b986e3a00db44 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A103633 → basis A013611