A143683 → 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, 12,   12;
  3: 1, 22,   60,    40;
  4: 1, 32,  208,   352,    176;
  5: 1, 42,  456,  1424,   1680,    672;
  6: 1, 52,  804,  4256,   9008,   8256,    2752;
  7: 1, 62, 1252,  9848,  32560,  51104,   38080,   10880;
  8: 1, 72, 1800, 19200,  92736, 222336,  278400,  175104,  43776;
  9: 1, 82, 2448, 33312, 219936, 747072, 1392384, 1445376, 785664, 174592;

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (10*n^2-27*n+18+(76*n^2-233*n+162)*x+(240*n^2-771*n+540)*x^2+(400*n^2-1293*n+900)*x^3+(370*n^2-1178*n+810)*x^4+(180*n^2-558*n+378)*x^5+(36*n^2-108*n+72)*x^6)*P(n-1,x) - (n-1+(42*n-95)*x+(266*n-604)*x^2+(736*n-1630)*x^3+(1089*n-2345)*x^4+(902*n-1891)*x^5+(396*n-810)*x^6+(72*n-144)*x^7)*d/dx P(n-1,x) + (x+41*x^2+226*x^3+550*x^4+725*x^5+541*x^6+216*x^7+36*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[((10*n^2 - 27*n + 18 + (76*n^2 - 233*n + 162)*x + (240*n^2 - 771*n + 540)*x^2 + (400*n^2 - 1293*n + 900)*x^3 + (370*n^2 - 1178*n + 810)*x^4 + (180*n^2 - 558*n + 378)*x^5 + (36*n^2 - 108*n + 72)*x^6)*P[n - 1, x] - (n - 1 + (42*n - 95)*x + (266*n - 604)*x^2 + (736*n - 1630)*x^3 + (1089*n - 2345)*x^4 + (902*n - 1891)*x^5 + (396*n - 810)*x^6 + (72*n - 144)*x^7)*D[P[n - 1, x], x] + (x + 41*x^2 + 226*x^3 + 550*x^4 + 725*x^5 + 541*x^6 + 216*x^7 + 36*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] = ((10*n^2 - 27*n + 18 + (76*n^2 - 233*n + 162)*x + (240*n^2 - 771*n + 540)*x^2 + (400*n^2 - 1293*n + 900)*x^3 + (370*n^2 - 1178*n + 810)*x^4 + (180*n^2 - 558*n + 378)*x^5 + (36*n^2 - 108*n + 72)*x^6)*v[i - 1] - (n - 1 + (42*n - 95)*x + (266*n - 604)*x^2 + (736*n - 1630)*x^3 + (1089*n - 2345)*x^4 + (902*n - 1891)*x^5 + (396*n - 810)*x^6 + (72*n - 144)*x^7)*deriv(v[i - 1], x) + (x + 41*x^2 + 226*x^3 + 550*x^4 + 725*x^5 + 541*x^6 + 216*x^7 + 36*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-6f46dc63d04bd23c
• 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-6f46dc63d04bd23c
• 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-6f46dc63d04bd23c

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-6f46dc63d04bd23c

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Pascal-(1,8,1) 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-a922c934805482be root: theorem check

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

    • preserver_image reversal, shift 0 e-e737400fb7cbaf29

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,8,1) 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-a922c934805482be (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A143683 → 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