basis-map coefficients of A026729 → basis A056242 → D+1

OEIS matches ?

• A181331: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  1;
   2:  3,    2;
   3:  6,   13,    4;
   4: 10,   41,   40,     8;
   5: 15,   96,  173,   108,    16;
   6: 21,  190,  523,   594,   272,     32;
   7: 28,  337, 1286,  2201,  1816,    656,    64;
   8: 36,  553, 2758,  6465,  7968,   5152,  1536,   128;
   9: 45,  856, 5362, 16248, 27209,  26120, 13872,  3520,  256;
  10: 55, 1266, 9678, 36468, 78371, 101650, 79792, 35936, 7936, 512;

The row polynomials satisfy (n^3-2*n^2+n+(6*n-6)*x-(4*n^2-12*n+8)*x^2-(4*n-4)*x^3)*P(n,x) = (n^3-n+(6*n^2-2)*x-(4*n^2-20*n+12)*x^2-(12*n-12)*x^3-8*x^4)*P(n-1,x) + ((2*n^2-2*n)*x+(2*n^2-2*n-4)*x^2-(8*n+4)*x^3-(8*n+8)*x^4-8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - n + (6*n^2 - 2)*x - (4*n^2 - 20*n + 12)*x^2 - (12*n - 12)*x^3 - 8*x^4)*P[n - 1, x] + ((2*n^2 - 2*n)*x + (2*n^2 - 2*n - 4)*x^2 - (8*n + 4)*x^3 - (8*n + 8)*x^4 - 8*x^5)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n + (6*n - 6)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n - 4)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^3 - n + (6*n^2 - 2)*x - (4*n^2 - 20*n + 12)*x^2 - (12*n - 12)*x^3 - 8*x^4)*v[i - 1] + ((2*n^2 - 2*n)*x + (2*n^2 - 2*n - 4)*x^2 - (8*n + 4)*x^3 - (8*n + 8)*x^4 - 8*x^5)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n + (6*n - 6)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n - 4)*x^3)); 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-a682124e3a957353
• 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-a682124e3a957353
• 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-a682124e3a957353

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-a682124e3a957353

  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.