A059473 → 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:    4,     2;
  2:   14,    18,      4;
  3:   40,    96,     56,      8;
  4:  104,   384,    412,    152,      16;
  5:  256,  1296,   2144,   1432,     384,     32;
  6:  608,  3936,   9016,   9224,    4416,    928,     64;
  7: 1408, 11136,  32960,  46720,   33984,  12608,   2176,   128;
  8: 3200, 29952, 109312, 200960,  201936, 112960,  34112,  4992,   256;
  9: 7168, 77568, 337664, 769024, 1002432, 771488, 348928, 88704, 11264, 512;

The row polynomials satisfy -(3*n^3-5*n^2+2*n-(3*n^3-5*n^2+14*n)*x-(4*n^2-8*n)*x^2-2*n^2*x^3-2*n*x^4)*P(n,x) = -(6*n^3-4*n^2-2*n+(12*n^3-26*n^2-34*n)*x+(10*n^2-54*n+36)*x^2+(8*n^2-4*n-4)*x^3+(4*n-16)*x^4-4*x^5)*P(n-1,x) + ((6*n^2+2*n)*x+(24*n^2+20*n-4)*x^2+(6*n^2+54*n-4)*x^3+(28*n+48)*x^4+(4*n+28)*x^5+4*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[(-(6*n^3 - 4*n^2 - 2*n + (12*n^3 - 26*n^2 - 34*n)*x + (10*n^2 - 54*n + 36)*x^2 + (8*n^2 - 4*n - 4)*x^3 + (4*n - 16)*x^4 - 4*x^5)*P[n - 1, x] + ((6*n^2 + 2*n)*x + (24*n^2 + 20*n - 4)*x^2 + (6*n^2 + 54*n - 4)*x^3 + (28*n + 48)*x^4 + (4*n + 28)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-3*n^3 + 5*n^2 - 2*n + (3*n^3 - 5*n^2 + 14*n)*x + (4*n^2 - 8*n)*x^2 + 2*n^2*x^3 + 2*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] = (-(6*n^3 - 4*n^2 - 2*n + (12*n^3 - 26*n^2 - 34*n)*x + (10*n^2 - 54*n + 36)*x^2 + (8*n^2 - 4*n - 4)*x^3 + (4*n - 16)*x^4 - 4*x^5)*v[i - 1] + ((6*n^2 + 2*n)*x + (24*n^2 + 20*n - 4)*x^2 + (6*n^2 + 54*n - 4)*x^3 + (28*n + 48)*x^4 + (4*n + 28)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-3*n^3 + 5*n^2 - 2*n + (3*n^3 - 5*n^2 + 14*n)*x + (4*n^2 - 8*n)*x^2 + 2*n^2*x^3 + 2*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-41032588237d8238
• 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-41032588237d8238
• 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-41032588237d8238
• 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-41032588237d8238

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-41032588237d8238

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

  • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
    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-0ee228386582da48 root: theorem check

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

    • preserver_image reversal, shift 0 e-c68c82cb1005adf9

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, .…
        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-0ee228386582da48 (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.