A144438 → D

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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:    0;
  1:    1;
  2:    5,      2;
  3:   14,     28,       3;
  4:   33,    178,      99,       4;
  5:   72,    826,    1239,     288,       5;
  6:  151,   3264,   10179,    6528,     755,       6;
  7:  310,  11748,   66435,   88580,   29370,    1860,      7;
  8:  629,  39886,  376368,  899372,  627280,  119658,   4403,     8;
  9: 1268, 130342, 1941897, 7596384, 9495480, 3883794, 456197, 10144, 9;

The row polynomials satisfy n*P(n,x) = -(n+1)*x^2*P(n-4,x) - ((2*n+2)*x+(2*n^2-2*n-3)*x^2)*P(n-3,x) - ((n+1)*x^2-(n+1)*x^3)*d/dx P(n-3,x) - (2*n+2+(2*n^2-4*n-4)*x+(n^3-2*n^2)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-n-1)*x^2-n^2*x^3)*d/dx P(n-2,x) + (3*n+1+(2*n^2-2*n)*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 14 + 28*x + 3*x^2.

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[3, x_] := 14 + 28*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1)*x^2*P[n - 4, x] - ((2*n + 2)*x + (2*n^2 - 2*n - 3)*x^2)*P[n - 3, x] - ((n + 1)*x^2 - (n + 1)*x^3)*D[P[n - 3, x], x] - (2*n + 2 + (2*n^2 - 4*n - 4)*x + (n^3 - 2*n^2)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - n - 1)*x^2 - n^2*x^3)*D[P[n - 2, x], x] + (3*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 14 + 28*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n + 1)*x^2*v[i - 4] - ((2*n + 2)*x + (2*n^2 - 2*n - 3)*x^2)*v[i - 3] - ((n + 1)*x^2 - (n + 1)*x^3)*deriv(v[i - 3], x) - (2*n + 2 + (2*n^2 - 4*n - 4)*x + (n^3 - 2*n^2)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - n - 1)*x^2 - n^2*x^3)*deriv(v[i - 2], x) + (3*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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-efbd5175f6e98006
• 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-efbd5175f6e98006
• 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-efbd5175f6e98006
• 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-efbd5175f6e98006

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-efbd5175f6e98006

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

  • premise (sequence): Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
    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-d401dbe9093b268d root: theorem check

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

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.