A116412 → x ↦ x/(x+1)

OEIS matches ?

• A124610: 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:
  0:   1;
  1:   3,     4;
  2:   6,    18,    13;
  3:  12,    57,    87,     43;
  4:  24,   156,   369,    378,    142;
  5:  48,   396,  1275,   2007,   1548,     469;
  6:  96,   960,  3918,   8364,   9864,    6102,    1549;
  7: 192,  2256, 11160,  30153,  48102,   45351,   23421,   5116;
  8: 384,  5184, 30144,  98664, 198945,  253236,  198861,  88146,  16897;
  9: 768, 11712, 78288, 301176, 735195, 1181562, 1250883, 841647, 326769, 55807;

The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (7*n^2-21*n+20+(44*n^2-146*n+128)*x+(119*n^2-414*n+337)*x^2+(176*n^2-615*n+469)*x^3+(149*n^2-506*n+365)*x^4+(68*n^2-219*n+151)*x^5+(13*n^2-39*n+26)*x^6)*P(n-1,x) - (4*n-8+(40*n-92)*x+(165*n-393)*x^2+(366*n-864)*x^3+(474*n-1086)*x^4+(360*n-792)*x^5+(149*n-313)*x^6+(26*n-52)*x^7)*d/dx P(n-1,x) + (4*x+36*x^2+133*x^3+265*x^4+310*x^5+214*x^6+81*x^7+13*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[((7*n^2 - 21*n + 20 + (44*n^2 - 146*n + 128)*x + (119*n^2 - 414*n + 337)*x^2 + (176*n^2 - 615*n + 469)*x^3 + (149*n^2 - 506*n + 365)*x^4 + (68*n^2 - 219*n + 151)*x^5 + (13*n^2 - 39*n + 26)*x^6)*P[n - 1, x] - (4*n - 8 + (40*n - 92)*x + (165*n - 393)*x^2 + (366*n - 864)*x^3 + (474*n - 1086)*x^4 + (360*n - 792)*x^5 + (149*n - 313)*x^6 + (26*n - 52)*x^7)*D[P[n - 1, x], x] + (4*x + 36*x^2 + 133*x^3 + 265*x^4 + 310*x^5 + 214*x^6 + 81*x^7 + 13*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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] = ((7*n^2 - 21*n + 20 + (44*n^2 - 146*n + 128)*x + (119*n^2 - 414*n + 337)*x^2 + (176*n^2 - 615*n + 469)*x^3 + (149*n^2 - 506*n + 365)*x^4 + (68*n^2 - 219*n + 151)*x^5 + (13*n^2 - 39*n + 26)*x^6)*v[i - 1] - (4*n - 8 + (40*n - 92)*x + (165*n - 393)*x^2 + (366*n - 864)*x^3 + (474*n - 1086)*x^4 + (360*n - 792)*x^5 + (149*n - 313)*x^6 + (26*n - 52)*x^7)*deriv(v[i - 1], x) + (4*x + 36*x^2 + 133*x^3 + 265*x^4 + 310*x^5 + 214*x^6 + 81*x^7 + 13*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*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-a802d54c8a2137b8
• 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-a802d54c8a2137b8
• 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-a802d54c8a2137b8

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-a802d54c8a2137b8

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
    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-ee40b91a84fdd5b7 root: theorem check

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

    • preserver_image reversal, shift 0 e-c36241f3b645d7b5

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
        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-67b682ca6511d839 root: theorem check

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

        • preserver_image reversal, shift 0 e-c7b023a4533e2987

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
            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-ee40b91a84fdd5b7 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A249251 → 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