A103633 → 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: 1, 1;
  2: 1, 3, 1;
  3: 0, 2, 4,  1;
  4: 0, 2, 7,  6,  1;
  5: 0, 0, 3,  9,  7,  1;
  6: 0, 0, 3, 13, 18,  9,  1;
  7: 0, 0, 0,  4, 16, 21, 10,  1;
  8: 0, 0, 0,  4, 21, 40, 34, 12,  1;
  9: 0, 0, 0,  0,  5, 25, 46, 38, 13, 1;

The row polynomials satisfy (n^2-n+(4*n^2-6*n)*x+(4*n^2-10)*x^2+(8*n-8)*x^3+4*x^4)*P(n,x) = -(n^3-4*n^2+3*n+(6*n^3-19*n^2+9*n+6)*x+(12*n^3-28*n^2+24)*x^2+(8*n^3-4*n^2-28*n+30)*x^3+(16*n^2-28*n+8)*x^4+(8*n-12)*x^5)*P(n-1,x) + ((2*n^2-2*n)*x+(10*n^2-6*n-4)*x^2+(16*n^2+4*n-12)*x^3+(8*n^2+24*n-8)*x^4+(16*n+8)*x^5+8*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[(-(n^3 - 4*n^2 + 3*n + (6*n^3 - 19*n^2 + 9*n + 6)*x + (12*n^3 - 28*n^2 + 24)*x^2 + (8*n^3 - 4*n^2 - 28*n + 30)*x^3 + (16*n^2 - 28*n + 8)*x^4 + (8*n - 12)*x^5)*P[n - 1, x] + ((2*n^2 - 2*n)*x + (10*n^2 - 6*n - 4)*x^2 + (16*n^2 + 4*n - 12)*x^3 + (8*n^2 + 24*n - 8)*x^4 + (16*n + 8)*x^5 + 8*x^6)*D[P[n - 1, x], x])/(n^2 - n + (4*n^2 - 6*n)*x + (4*n^2 - 10)*x^2 + (8*n - 8)*x^3 + 4*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] = (-(n^3 - 4*n^2 + 3*n + (6*n^3 - 19*n^2 + 9*n + 6)*x + (12*n^3 - 28*n^2 + 24)*x^2 + (8*n^3 - 4*n^2 - 28*n + 30)*x^3 + (16*n^2 - 28*n + 8)*x^4 + (8*n - 12)*x^5)*v[i - 1] + ((2*n^2 - 2*n)*x + (10*n^2 - 6*n - 4)*x^2 + (16*n^2 + 4*n - 12)*x^3 + (8*n^2 + 24*n - 8)*x^4 + (16*n + 8)*x^5 + 8*x^6)*deriv(v[i - 1], x))/(n^2 - n + (4*n^2 - 6*n)*x + (4*n^2 - 10)*x^2 + (8*n - 8)*x^3 + 4*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-6059e077b07429fb
• 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-6059e077b07429fb
• 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-6059e077b07429fb

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-6059e077b07429fb

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

  • premise (sequence): Triangle read by rows: triangle of repeated stepped binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-35a33ce2a6b89415 root: theorem check

      Parent-row provers of polytool polytool ?

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.