A113413 → x ↦ x/(x+1)

OEIS matches ?

• A007483: 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: 2,  3;
  2: 2,  8,   7;
  3: 2, 14,  28,   17;
  4: 2, 20,  66,   88,    41;
  5: 2, 26, 122,  262,   262,    99;
  6: 2, 32, 196,  592,   940,   752,   239;
  7: 2, 38, 288, 1132,  2508,  3156,  2104,   577;
  8: 2, 44, 398, 1936,  5560,  9704, 10108,  5776,  1393;
  9: 2, 50, 526, 3058, 10852, 24464, 35180, 31252, 15626, 3363;

The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (3*n^2-10*n+11+(20*n^2-75*n+73)*x+(58*n^2-225*n+197)*x^2+(92*n^2-349*n+279)*x^3+(83*n^2-297*n+220)*x^4+(40*n^2-132*n+92)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n-2+(14*n-37)*x+(70*n-188)*x^2+(176*n-454)*x^3+(249*n-606)*x^4+(202*n-461)*x^5+(88*n-188)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 10*n + 11 + (20*n^2 - 75*n + 73)*x + (58*n^2 - 225*n + 197)*x^2 + (92*n^2 - 349*n + 279)*x^3 + (83*n^2 - 297*n + 220)*x^4 + (40*n^2 - 132*n + 92)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 37)*x + (70*n - 188)*x^2 + (176*n - 454)*x^3 + (249*n - 606)*x^4 + (202*n - 461)*x^5 + (88*n - 188)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*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; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 10*n + 11 + (20*n^2 - 75*n + 73)*x + (58*n^2 - 225*n + 197)*x^2 + (92*n^2 - 349*n + 279)*x^3 + (83*n^2 - 297*n + 220)*x^4 + (40*n^2 - 132*n + 92)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n - 2 + (14*n - 37)*x + (70*n - 188)*x^2 + (176*n - 454)*x^3 + (249*n - 606)*x^4 + (202*n - 461)*x^5 + (88*n - 188)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*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-73b4fdfae3fbc1a0
• 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-73b4fdfae3fbc1a0
• 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-73b4fdfae3fbc1a0

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-73b4fdfae3fbc1a0

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): A Riordan array of coordination sequences.
    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-7fe1799d89540b32 root: theorem check

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

Related sequences ?

Same fingerprint (3)

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

• A122542 → x ↦ x/(x+1)
• A035607 → basis A007318
• A035607 → x ↦ x+1

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