A008277 → x ↦ x/(x+1)

OEIS matches ?

• A004211: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A059340: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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,   2;
   3: 1,   5,     5;
   4: 1,  10,    23,     15;
   5: 1,  19,    76,    109,     52;
   6: 1,  36,   224,    531,    544,     203;
   7: 1,  69,   631,   2204,   3641,    2876,     877;
   8: 1, 134,  1749,   8471,  20089,   25208,   16113,    4140;
   9: 1, 263,  4838,  31331, 100171,  177631,  178564,   95495,  21147;
  10: 1, 520, 13454, 113807, 471827, 1109473, 1559788, 1302451, 597155, 115975;

The row polynomials satisfy (1+x)*P(n,x) = (1-(n-5)*x-(2*n-6)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
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] = ((1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); 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)

Triangle begins:
   1:      1;
   2:      2,      1;
   3:      5,      5,       1;
   4:     15,     23,      10,       1;
   5:     52,    109,      76,      19,       1;
   6:    203,    544,     531,     224,      36,      1;
   7:    877,   2876,    3641,    2204,     631,     69,      1;
   8:   4140,  16113,   25208,   20089,    8471,   1749,    134,     1;
   9:  21147,  95495,  178564,  177631,  100171,  31331,   4838,   263,   1;
  10: 115975, 597155, 1302451, 1559788, 1109473, 471827, 113807, 13454, 520, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where (1+x)*P(n,x) = (1-(n-5)*x-(2*n-6)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[((1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 - (n - 5)*x - (2*n - 6)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); 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-6e9c730f2a536c2a
• 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-6e9c730f2a536c2a
• 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-6e9c730f2a536c2a

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-6e9c730f2a536c2a

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-995393ce4e49a8d6 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • basis_map A008277 e-c52960bc5e97caa0

      Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_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 (-inf, 0]
        • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-ad67990681a86223

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • 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 (-inf, 0]
            • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)
    • preserver_image reversal, shift 0 e-d527982dc7c0aa3b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-8e580efc86a38a12 root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • preserver_image reversal, shift 0 e-3256729839b26765

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang e-995393ce4e49a8d6 (expanded above)
            • ↵ basis_map A008277 e-c52960bc5e97caa0 (expanded above)

Related sequences ?

Same fingerprint (5)

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

• A008278 → x ↦ x+1
• A106800 → reverse (shift +1) → x ↦ x/(x+1)
• A008278 → x ↦ x+1 (shift +1) → reverse
• A106800 → x ↦ x+1
• A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741