basis-map coefficients of A026729 → basis A056242 → D

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:
   1:  0;
   2:  2;
   3:  5,    8;
   4:  9,   32,   24;
   5: 14,   82,  132,    64;
   6: 20,  170,  438,   448,   160;
   7: 27,  310, 1131,  1824,  1360,   384;
   8: 35,  518, 2499,  5632,  6560,  3840,   896;
   9: 44,  812, 4956, 14596, 23560, 21408, 10304,  2048;
  10: 54, 1212, 9072, 33444, 70010, 87648, 65184, 26624, 4608;

The row polynomials satisfy (n^3-2*n^2-n+2+(4*n^2-12*n+8)*x)*P(n,x) = (n^3-3*n+2+(10*n^2-18*n+16)*x+(16*n-16)*x^2)*P(n-1,x) + ((2*n^2-2*n+4)*x+(2*n^2+6*n-4)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n + 2 + (10*n^2 - 18*n + 16)*x + (16*n - 16)*x^2)*P[n - 1, x] + ((2*n^2 - 2*n + 4)*x + (2*n^2 + 6*n - 4)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(n^3 - 2*n^2 - n + 2 + (4*n^2 - 12*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n^3 - 3*n + 2 + (10*n^2 - 18*n + 16)*x + (16*n - 16)*x^2)*v[i - 1] + ((2*n^2 - 2*n + 4)*x + (2*n^2 + 6*n - 4)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(n^3 - 2*n^2 - n + 2 + (4*n^2 - 12*n + 8)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-1da1d1636f7486ed
• Coefficients of one sign. Every nonzero row, n >= 1: 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-1da1d1636f7486ed
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-1da1d1636f7486ed

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-1da1d1636f7486ed

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

  • premise (sequence): Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A056242 e-3cb5f3d64c4bb317

      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
        no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

1. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.