A249251 → x ↦ x/(x+1)

OEIS matches ?

• A123347: 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,  8,  13;
  3: 1, 12,  42,   43;
  4: 1, 16,  87,  190,   142;
  5: 1, 20, 148,  505,   797,   469;
  6: 1, 24, 225, 1052,  2589,  3192,   1549;
  7: 1, 28, 318, 1895,  6398, 12261,  12391,   5116;
  8: 1, 32, 427, 3098, 13360, 35096,  54955,  47030,  16897;
  9: 1, 36, 552, 4725, 24867, 83469, 178902, 236547, 175494, 55807;

The row polynomials satisfy (2*n-2+(6*n-6)*x+(6*n-6)*x^2+(2*n-2)*x^3)*P(n,x) = (4*n^2-2*n-2+(26*n^2-32*n+6)*x+(77*n^2-151*n+74)*x^2+(128*n^2-320*n+192)*x^3+(122*n^2-344*n+222)*x^4+(62*n^2-184*n+122)*x^5+(13*n^2-39*n+26)*x^6)*P(n-1,x) - (n+(16*n-16)*x+(87*n-128)*x^2+(234*n-404)*x^3+(351*n-656)*x^4+(300*n-584)*x^5+(137*n-272)*x^6+(26*n-52)*x^7)*d/dx P(n-1,x) + (x+15*x^2+73*x^3+175*x^4+235*x^5+181*x^6+75*x^7+13*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n - 2 + (26*n^2 - 32*n + 6)*x + (77*n^2 - 151*n + 74)*x^2 + (128*n^2 - 320*n + 192)*x^3 + (122*n^2 - 344*n + 222)*x^4 + (62*n^2 - 184*n + 122)*x^5 + (13*n^2 - 39*n + 26)*x^6)*P[n - 1, x] - (n + (16*n - 16)*x + (87*n - 128)*x^2 + (234*n - 404)*x^3 + (351*n - 656)*x^4 + (300*n - 584)*x^5 + (137*n - 272)*x^6 + (26*n - 52)*x^7)*D[P[n - 1, x], x] + (x + 15*x^2 + 73*x^3 + 175*x^4 + 235*x^5 + 181*x^6 + 75*x^7 + 13*x^8)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n - 2 + (26*n^2 - 32*n + 6)*x + (77*n^2 - 151*n + 74)*x^2 + (128*n^2 - 320*n + 192)*x^3 + (122*n^2 - 344*n + 222)*x^4 + (62*n^2 - 184*n + 122)*x^5 + (13*n^2 - 39*n + 26)*x^6)*v[i - 1] - (n + (16*n - 16)*x + (87*n - 128)*x^2 + (234*n - 404)*x^3 + (351*n - 656)*x^4 + (300*n - 584)*x^5 + (137*n - 272)*x^6 + (26*n - 52)*x^7)*deriv(v[i - 1], x) + (x + 15*x^2 + 73*x^3 + 175*x^4 + 235*x^5 + 181*x^6 + 75*x^7 + 13*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (6*n - 6)*x + (6*n - 6)*x^2 + (2*n - 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-24dc52663a6481f0
• 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-24dc52663a6481f0
• 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-24dc52663a6481f0

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-24dc52663a6481f0

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-67b682ca6511d839 root: theorem check

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

    • preserver_image reversal, shift 0 e-c7b023a4533e2987

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
        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-ee40b91a84fdd5b7 root: theorem check

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

        • preserver_image reversal, shift 0 e-c36241f3b645d7b5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
            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-67b682ca6511d839 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A116412 → basis A007318

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