A081579 → 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:  7,    8,     1;
  3: 12,   33,    14,     1;
  4: 17,  108,    94,    20,     1;
  5: 22,  233,   424,   190,    26,     1;
  6: 27,  408,  1379,  1160,   321,    32,    1;
  7: 32,  633,  3334,  5055,  2516,   487,   38,   1;
  8: 37,  908,  6664, 16500, 13986,  4692,  688,  44,  1;
  9: 42, 1233, 11744, 42620, 58356, 32022, 7888, 924, 50, 1;

The row polynomials satisfy -(5*n^3-13*n^2+8*n-(5*n^3-20*n^2+42*n)*x-(16*n^2-39*n)*x^2-(n^2+14*n)*x^3-n*x^4)*P(n,x) = -(5*n^3-8*n^2+3*n+(45*n^3-150*n^2+51*n)*x+(139*n^2-353*n+233)*x^2+(9*n^2+111*n-127)*x^3+(8*n-25)*x^4-x^5)*P(n-1,x) + ((5*n^2-3*n)*x+(90*n^2-37*n-8)*x^2+(5*n^2+304*n-127)*x^3+(35*n+299)*x^4+(n+35)*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[(-(5*n^3 - 8*n^2 + 3*n + (45*n^3 - 150*n^2 + 51*n)*x + (139*n^2 - 353*n + 233)*x^2 + (9*n^2 + 111*n - 127)*x^3 + (8*n - 25)*x^4 - x^5)*P[n - 1, x] + ((5*n^2 - 3*n)*x + (90*n^2 - 37*n - 8)*x^2 + (5*n^2 + 304*n - 127)*x^3 + (35*n + 299)*x^4 + (n + 35)*x^5 + x^6)*D[P[n - 1, x], x])/(-5*n^3 + 13*n^2 - 8*n + (5*n^3 - 20*n^2 + 42*n)*x + (16*n^2 - 39*n)*x^2 + (n^2 + 14*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] = (-(5*n^3 - 8*n^2 + 3*n + (45*n^3 - 150*n^2 + 51*n)*x + (139*n^2 - 353*n + 233)*x^2 + (9*n^2 + 111*n - 127)*x^3 + (8*n - 25)*x^4 - x^5)*v[i - 1] + ((5*n^2 - 3*n)*x + (90*n^2 - 37*n - 8)*x^2 + (5*n^2 + 304*n - 127)*x^3 + (35*n + 299)*x^4 + (n + 35)*x^5 + x^6)*deriv(v[i - 1], x))/(-5*n^3 + 13*n^2 - 8*n + (5*n^3 - 20*n^2 + 42*n)*x + (16*n^2 - 39*n)*x^2 + (n^2 + 14*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-ca8ef6d6c750b1d1
• 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-ca8ef6d6c750b1d1
• 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-ca8ef6d6c750b1d1
• 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-ca8ef6d6c750b1d1

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-ca8ef6d6c750b1d1

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

  • premise (sequence): Pascal-(1,4,1) 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-01acb0cb09e95244 root: theorem check

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

    • preserver_image reversal, shift 0 e-93c62bb8c1382161

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,4,1) 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-01acb0cb09e95244 (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.