A081582 → 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, 11,   11;
  3: 1, 20,   54,    36;
  4: 1, 29,  178,   298,    149;
  5: 1, 38,  383,  1172,   1375,    550;
  6: 1, 47,  669,  3387,   7051,   6429,   2143;
  7: 1, 56, 1036,  7672,  24710,  38360,  28476,   8136;
  8: 1, 65, 1484, 14756,  68446, 160846, 199556, 125092,  31273;
  9: 1, 74, 2013, 25368, 158914, 524412, 962738, 992056, 537741, 119498;

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (9*n^2-24*n+16+(68*n^2-207*n+144)*x+(214*n^2-685*n+480)*x^2+(356*n^2-1149*n+800)*x^3+(329*n^2-1047*n+720)*x^4+(160*n^2-496*n+336)*x^5+(32*n^2-96*n+64)*x^6)*P(n-1,x) - (n-1+(38*n-85)*x+(238*n-538)*x^2+(656*n-1450)*x^3+(969*n-2085)*x^4+(802*n-1681)*x^5+(352*n-720)*x^6+(64*n-128)*x^7)*d/dx P(n-1,x) + (x+37*x^2+202*x^3+490*x^4+645*x^5+481*x^6+192*x^7+32*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[((9*n^2 - 24*n + 16 + (68*n^2 - 207*n + 144)*x + (214*n^2 - 685*n + 480)*x^2 + (356*n^2 - 1149*n + 800)*x^3 + (329*n^2 - 1047*n + 720)*x^4 + (160*n^2 - 496*n + 336)*x^5 + (32*n^2 - 96*n + 64)*x^6)*P[n - 1, x] - (n - 1 + (38*n - 85)*x + (238*n - 538)*x^2 + (656*n - 1450)*x^3 + (969*n - 2085)*x^4 + (802*n - 1681)*x^5 + (352*n - 720)*x^6 + (64*n - 128)*x^7)*D[P[n - 1, x], x] + (x + 37*x^2 + 202*x^3 + 490*x^4 + 645*x^5 + 481*x^6 + 192*x^7 + 32*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] = ((9*n^2 - 24*n + 16 + (68*n^2 - 207*n + 144)*x + (214*n^2 - 685*n + 480)*x^2 + (356*n^2 - 1149*n + 800)*x^3 + (329*n^2 - 1047*n + 720)*x^4 + (160*n^2 - 496*n + 336)*x^5 + (32*n^2 - 96*n + 64)*x^6)*v[i - 1] - (n - 1 + (38*n - 85)*x + (238*n - 538)*x^2 + (656*n - 1450)*x^3 + (969*n - 2085)*x^4 + (802*n - 1681)*x^5 + (352*n - 720)*x^6 + (64*n - 128)*x^7)*deriv(v[i - 1], x) + (x + 37*x^2 + 202*x^3 + 490*x^4 + 645*x^5 + 481*x^6 + 192*x^7 + 32*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-73cab1ceabab0270
• 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-73cab1ceabab0270
• 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-73cab1ceabab0270

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-73cab1ceabab0270

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Pascal-(1,7,1) 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-a31f5c33917b15a7 root: theorem check

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

    • preserver_image reversal, shift 0 e-7d90ede2da8763e3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,7,1) 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-a31f5c33917b15a7 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A081582 → 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