A081577 → 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:  5,   6,    1;
  3:  8,  21,   10,    1;
  4: 11,  54,   52,   14,     1;
  5: 14, 105,  184,   98,    18,    1;
  6: 17, 174,  487,  452,   159,   22,    1;
  7: 20, 261, 1042, 1535,   912,  235,   26,   1;
  8: 23, 366, 1930, 4130,  3816, 1618,  326,  30,  1;
  9: 26, 489, 3232, 9344, 12516, 8086, 2624, 432, 34, 1;

The row polynomials satisfy -(3*n^3-7*n^2+4*n-(3*n^3-10*n^2+22*n)*x-(8*n^2-19*n)*x^2-(n^2+6*n)*x^3-n*x^4)*P(n,x) = -(3*n^3-4*n^2+n+(15*n^3-44*n^2-n)*x+(37*n^2-99*n+69)*x^2+(5*n^2+23*n-31)*x^3+(4*n-13)*x^4-x^5)*P(n-1,x) + ((3*n^2-n)*x+(30*n^2-n-4)*x^2+(3*n^2+90*n-31)*x^3+(19*n+87)*x^4+(n+19)*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[(-(3*n^3 - 4*n^2 + n + (15*n^3 - 44*n^2 - n)*x + (37*n^2 - 99*n + 69)*x^2 + (5*n^2 + 23*n - 31)*x^3 + (4*n - 13)*x^4 - x^5)*P[n - 1, x] + ((3*n^2 - n)*x + (30*n^2 - n - 4)*x^2 + (3*n^2 + 90*n - 31)*x^3 + (19*n + 87)*x^4 + (n + 19)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 + 7*n^2 - 4*n + (3*n^3 - 10*n^2 + 22*n)*x + (8*n^2 - 19*n)*x^2 + (n^2 + 6*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] = (-(3*n^3 - 4*n^2 + n + (15*n^3 - 44*n^2 - n)*x + (37*n^2 - 99*n + 69)*x^2 + (5*n^2 + 23*n - 31)*x^3 + (4*n - 13)*x^4 - x^5)*v[i - 1] + ((3*n^2 - n)*x + (30*n^2 - n - 4)*x^2 + (3*n^2 + 90*n - 31)*x^3 + (19*n + 87)*x^4 + (n + 19)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 + 7*n^2 - 4*n + (3*n^3 - 10*n^2 + 22*n)*x + (8*n^2 - 19*n)*x^2 + (n^2 + 6*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-303886c645549307
• 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-303886c645549307
• 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-303886c645549307
• 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-303886c645549307

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-303886c645549307

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

  • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
    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-1f8dc4b132ba6116 root: theorem check

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

    • preserver_image reversal, shift 0 e-8efc9aafb30c2198

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,2,1) array read by antidiagonals.
        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-1f8dc4b132ba6116 (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.