A100862 → x ↦ x/(x+1)

OEIS matches ?

• A083886: 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,  2;
  2: 1,  5,    5;
  3: 1,  9,   21,   14;
  4: 1, 14,   57,   86,    43;
  5: 1, 20,  125,  320,   355,    142;
  6: 1, 27,  240,  925,  1710,   1497,    499;
  7: 1, 35,  420, 2275,  6230,   8967,   6475,   1850;
  8: 1, 44,  686, 4984, 18935,  39872,  46858,  28772,   7193;
  9: 1, 54, 1062, 9996, 50463, 145782, 248178, 246024, 131337, 29186;

The row polynomials satisfy (2+4*x+2*x^2)*P(n,x) = (n^2+n+(7*n^2-11*n+12)*x+(19*n^2-51*n+42)*x^2+(25*n^2-77*n+56)*x^3+(16*n^2-50*n+34)*x^4+(4*n^2-12*n+8)*x^5)*P(n-1,x) - (2+4*n*x+(24*n-38)*x^2+(56*n-110)*x^3+(64*n-132)*x^4+(36*n-74)*x^5+(8*n-16)*x^6)*d/dx P(n-1,x) + (4*x^2+20*x^3+40*x^4+40*x^5+20*x^6+4*x^7)*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[((n^2 + n + (7*n^2 - 11*n + 12)*x + (19*n^2 - 51*n + 42)*x^2 + (25*n^2 - 77*n + 56)*x^3 + (16*n^2 - 50*n + 34)*x^4 + (4*n^2 - 12*n + 8)*x^5)*P[n - 1, x] - (2 + 4*n*x + (24*n - 38)*x^2 + (56*n - 110)*x^3 + (64*n - 132)*x^4 + (36*n - 74)*x^5 + (8*n - 16)*x^6)*D[P[n - 1, x], x] + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*x^7)*D[P[n - 1, x], {x, 2}])/(2 + 4*x + 2*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 + n + (7*n^2 - 11*n + 12)*x + (19*n^2 - 51*n + 42)*x^2 + (25*n^2 - 77*n + 56)*x^3 + (16*n^2 - 50*n + 34)*x^4 + (4*n^2 - 12*n + 8)*x^5)*v[i - 1] - (2 + 4*n*x + (24*n - 38)*x^2 + (56*n - 110)*x^3 + (64*n - 132)*x^4 + (36*n - 74)*x^5 + (8*n - 16)*x^6)*deriv(v[i - 1], x) + (4*x^2 + 20*x^3 + 40*x^4 + 40*x^5 + 20*x^6 + 4*x^7)*deriv(deriv(v[i - 1], x), x))/(2 + 4*x + 2*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-16a1260593e0941e
• 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-16a1260593e0941e
• 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-16a1260593e0941e

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-16a1260593e0941e

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
    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-e7942008ad3e1420 root: theorem check

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

    • preserver_image reversal, shift 0 e-7078061f6d32a158

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the compl…
        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-e7942008ad3e1420 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A100862 → 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