A114655 → x ↦ x/(x+1)

OEIS matches ?

• A086871: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    2;
   2:    4,     6;
   3:    8,    28,     22;
   4:   16,    96,    168,      90;
   5:   32,   288,    832,     968,     394;
   6:   64,   800,   3360,    6320,    5500,     1806;
   7:  128,  2112,  12000,   32240,   44760,    31092,     8558;
   8:  256,  5376,  39424,  141120,  276640,   303744,   175504,    41586;
   9:  512, 13312, 121856,  555520, 1436288,  2208640,  2003456,   990736,  206098;
  10: 1024, 32256, 359424, 2021376, 6596352, 13281408, 16770432, 12951648, 5597172, 1037718;

The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (2*n+2+(5*n+11)*x+(2*n+20)*x^2-(3*n-15)*x^3-(2*n-4)*x^4)*P(n-1,x) + (4*x+14*x^2+18*x^3+10*x^4+2*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (5*n + 11)*x + (2*n + 20)*x^2 - (3*n - 15)*x^3 - (2*n - 4)*x^4)*P[n - 1, x] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*x^5)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (5*n + 11)*x + (2*n + 20)*x^2 - (3*n - 15)*x^3 - (2*n - 4)*x^4)*v[i - 1] + (4*x + 14*x^2 + 18*x^3 + 10*x^4 + 2*x^5)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)); 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)

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-460d39e35063c243
• 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-460d39e35063c243
• 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-460d39e35063c243

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-460d39e35063c243

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schr…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-784ea0037cdd0731 root: theorem check

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

Related sequences ?

Same fingerprint (2)

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

• A114687 → basis A007318
• A114656 → x ↦ 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