A143685 → 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: 12,   13,     1;
  3: 22,   63,    24,      1;
  4: 32,  313,   234,     35,      1;
  5: 42,  763,  1444,    525,     46,      1;
  6: 52, 1413,  6654,   4715,    936,     57,     1;
  7: 62, 2263, 18864,  29605,  11426,   1467,    68,    1;
  8: 72, 3313, 41074, 132195,  95516,  22877,  2118,   79,  1;
  9: 82, 4563, 76284, 409485, 577206, 244287, 40368, 2889, 90, 1;

The row polynomials satisfy -(10*n^3-28*n^2+18*n-(10*n^3-45*n^2+92*n)*x-(36*n^2-89*n)*x^2-(n^2+34*n)*x^3-n*x^4)*P(n,x) = -(10*n^3-18*n^2+8*n+(190*n^3-695*n^2+391*n)*x+(674*n^2-1688*n+1063)*x^2+(19*n^2+611*n-647)*x^3+(18*n-55)*x^4-x^5)*P(n-1,x) + ((10*n^2-8*n)*x+(380*n^2-267*n-18)*x^2+(10*n^2+1399*n-647)*x^3+(75*n+1389)*x^4+(n+75)*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[(-(10*n^3 - 18*n^2 + 8*n + (190*n^3 - 695*n^2 + 391*n)*x + (674*n^2 - 1688*n + 1063)*x^2 + (19*n^2 + 611*n - 647)*x^3 + (18*n - 55)*x^4 - x^5)*P[n - 1, x] + ((10*n^2 - 8*n)*x + (380*n^2 - 267*n - 18)*x^2 + (10*n^2 + 1399*n - 647)*x^3 + (75*n + 1389)*x^4 + (n + 75)*x^5 + x^6)*D[P[n - 1, x], x])/(-10*n^3 + 28*n^2 - 18*n + (10*n^3 - 45*n^2 + 92*n)*x + (36*n^2 - 89*n)*x^2 + (n^2 + 34*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] = (-(10*n^3 - 18*n^2 + 8*n + (190*n^3 - 695*n^2 + 391*n)*x + (674*n^2 - 1688*n + 1063)*x^2 + (19*n^2 + 611*n - 647)*x^3 + (18*n - 55)*x^4 - x^5)*v[i - 1] + ((10*n^2 - 8*n)*x + (380*n^2 - 267*n - 18)*x^2 + (10*n^2 + 1399*n - 647)*x^3 + (75*n + 1389)*x^4 + (n + 75)*x^5 + x^6)*deriv(v[i - 1], x))/(-10*n^3 + 28*n^2 - 18*n + (10*n^3 - 45*n^2 + 92*n)*x + (36*n^2 - 89*n)*x^2 + (n^2 + 34*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-36e148c47d980dd4
• 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-36e148c47d980dd4
• 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-36e148c47d980dd4
• 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-36e148c47d980dd4

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-36e148c47d980dd4

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

  • premise (sequence): Pascal-(1,9,1) 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-f64c8958fa0af494 root: theorem check

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

    • preserver_image reversal, shift 0 e-a327bf7575a1329d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,9,1) 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-f64c8958fa0af494 (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.