A081581 → x ↦ x/(x+1)

OEIS matches ?

No OEIS matches.

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, 10,   10;
  3: 1, 18,   48,    32;
  4: 1, 26,  150,   248,    124;
  5: 1, 34,  316,   944,   1100,    440;
  6: 1, 42,  546,  2632,   5376,   4872,   1624;
  7: 1, 50,  840,  5824,  18200,  27888,  20608,   5888;
  8: 1, 58, 1198, 11032,  48916, 112336, 137872,  86080,  21520;
  9: 1, 66, 1620, 18768, 110964, 354312, 639072, 652800, 352656, 78368;

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (8*n^2-21*n+14+(60*n^2-181*n+126)*x+(188*n^2-599*n+420)*x^2+(312*n^2-1005*n+700)*x^3+(288*n^2-916*n+630)*x^4+(140*n^2-434*n+294)*x^5+(28*n^2-84*n+56)*x^6)*P(n-1,x) - (n-1+(34*n-75)*x+(210*n-472)*x^2+(576*n-1270)*x^3+(849*n-1825)*x^4+(702*n-1471)*x^5+(308*n-630)*x^6+(56*n-112)*x^7)*d/dx P(n-1,x) + (x+33*x^2+178*x^3+430*x^4+565*x^5+421*x^6+168*x^7+28*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[((8*n^2 - 21*n + 14 + (60*n^2 - 181*n + 126)*x + (188*n^2 - 599*n + 420)*x^2 + (312*n^2 - 1005*n + 700)*x^3 + (288*n^2 - 916*n + 630)*x^4 + (140*n^2 - 434*n + 294)*x^5 + (28*n^2 - 84*n + 56)*x^6)*P[n - 1, x] - (n - 1 + (34*n - 75)*x + (210*n - 472)*x^2 + (576*n - 1270)*x^3 + (849*n - 1825)*x^4 + (702*n - 1471)*x^5 + (308*n - 630)*x^6 + (56*n - 112)*x^7)*D[P[n - 1, x], x] + (x + 33*x^2 + 178*x^3 + 430*x^4 + 565*x^5 + 421*x^6 + 168*x^7 + 28*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*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] = ((8*n^2 - 21*n + 14 + (60*n^2 - 181*n + 126)*x + (188*n^2 - 599*n + 420)*x^2 + (312*n^2 - 1005*n + 700)*x^3 + (288*n^2 - 916*n + 630)*x^4 + (140*n^2 - 434*n + 294)*x^5 + (28*n^2 - 84*n + 56)*x^6)*v[i - 1] - (n - 1 + (34*n - 75)*x + (210*n - 472)*x^2 + (576*n - 1270)*x^3 + (849*n - 1825)*x^4 + (702*n - 1471)*x^5 + (308*n - 630)*x^6 + (56*n - 112)*x^7)*deriv(v[i - 1], x) + (x + 33*x^2 + 178*x^3 + 430*x^4 + 565*x^5 + 421*x^6 + 168*x^7 + 28*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*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-1a90f5d53428f521
• 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-1a90f5d53428f521
• 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-1a90f5d53428f521

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-1a90f5d53428f521

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Pascal-(1,6,1) array.
    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-dfce858c9d52dee3 root: theorem check

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

    • preserver_image reversal, shift 0 e-5c5351a0cb2ba1df

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,6,1) array.
        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-dfce858c9d52dee3 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A081581 → 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