A081578 → D+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:  2,   1;
  2:  6,   7,    1;
  3: 10,  27,   12,     1;
  4: 14,  79,   72,    17,     1;
  5: 18, 163,  292,   141,    22,     1;
  6: 22, 279,  864,   761,   234,    27,    1;
  7: 26, 427, 1980,  2965,  1598,   351,   32,   1;
  8: 30, 607, 3832,  8865,  7826,  2915,  492,  37,  1;
  9: 34, 819, 6612, 21597, 29286, 17319, 4824, 657, 42, 1;

The row polynomials satisfy -(4*n^3-10*n^2+6*n-(4*n^3-15*n^2+32*n)*x-(12*n^2-29*n)*x^2-(n^2+10*n)*x^3-n*x^4)*P(n,x) = -(4*n^3-6*n^2+2*n+(28*n^3-89*n^2+19*n)*x+(80*n^2-206*n+139)*x^2+(7*n^2+59*n-71)*x^3+(6*n-19)*x^4-x^5)*P(n-1,x) + ((4*n^2-2*n)*x+(56*n^2-15*n-6)*x^2+(4*n^2+181*n-71)*x^3+(27*n+177)*x^4+(n+27)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 6*n^2 + 2*n + (28*n^3 - 89*n^2 + 19*n)*x + (80*n^2 - 206*n + 139)*x^2 + (7*n^2 + 59*n - 71)*x^3 + (6*n - 19)*x^4 - x^5)*P[n - 1, x] + ((4*n^2 - 2*n)*x + (56*n^2 - 15*n - 6)*x^2 + (4*n^2 + 181*n - 71)*x^3 + (27*n + 177)*x^4 + (n + 27)*x^5 + x^6)*D[P[n - 1, x], x])/(-4*n^3 + 10*n^2 - 6*n + (4*n^3 - 15*n^2 + 32*n)*x + (12*n^2 - 29*n)*x^2 + (n^2 + 10*n)*x^3 + n*x^4)]];
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] = (-(4*n^3 - 6*n^2 + 2*n + (28*n^3 - 89*n^2 + 19*n)*x + (80*n^2 - 206*n + 139)*x^2 + (7*n^2 + 59*n - 71)*x^3 + (6*n - 19)*x^4 - x^5)*v[i - 1] + ((4*n^2 - 2*n)*x + (56*n^2 - 15*n - 6)*x^2 + (4*n^2 + 181*n - 71)*x^3 + (27*n + 177)*x^4 + (n + 27)*x^5 + x^6)*deriv(v[i - 1], x))/(-4*n^3 + 10*n^2 - 6*n + (4*n^3 - 15*n^2 + 32*n)*x + (12*n^2 - 29*n)*x^2 + (n^2 + 10*n)*x^3 + n*x^4)); 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 (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-4c6b05990c1dd2cb
• 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-4c6b05990c1dd2cb
• 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-4c6b05990c1dd2cb
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-4c6b05990c1dd2cb

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 derivative_polynomial, shift 0 e-4c6b05990c1dd2cb

  Laguerre’s theorem on differential operators [SP14] polytool ?

  • premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 root: theorem check

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

    • preserver_image reversal, shift 0 e-86685874cd5f1179

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,3,1) 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-fba968684828f914 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.