basis-map coefficients of A021012 → D+1

OEIS matches ?

• A155456: row sums, 15 terms agree (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: -1, -1;
   3:  0,  4,   2;
   4:  0,  0, -18, -6;
   5:  0,  0,   0, 96,   24;
   6:  0,  0,   0,  0, -600, -120;
   7:  0,  0,   0,  0,    0, 4320,    720;
   8:  0,  0,   0,  0,    0,    0, -35280,  -5040;
   9:  0,  0,   0,  0,    0,    0,      0, 322560,    40320;
  10:  0,  0,   0,  0,    0,    0,      0,      0, -3265920, -362880;

The row polynomials satisfy (n-2+x)*P(n,x) = -((n^2-2*n+1)*x+(n-1)*x^2)*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^2 - 2*n + 1)*x + (n - 1)*x^2)*P[n - 1, x])/(n - 2 + x)]];
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^2 - 2*n + 1)*x + (n - 1)*x^2)*v[i - 1])/(n - 2 + 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-224ed4e0bc1e1628
• 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-224ed4e0bc1e1628
• 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-224ed4e0bc1e1628

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-224ed4e0bc1e1628

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

  • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-51f2fe49bf45c5d7 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-bb5cb95fed86aa9c

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-51f2fe49bf45c5d7 (expanded above)

Related sequences ?

Same fingerprint (10)

Equal up to content, sign, reversal and a small index shift.

• A023531 → D → D+1
• basis-map coefficients of A026729 → D+1
• A023531 → (θ+1)² → D+1
• basis-map coefficients of A026729, shifted → D+1
• A130123 → D+1
• A023531 → D+1
• basis-map coefficients of A030528, shifted → D+1
• A010054 → reverse → D+1
• A010054 → reverse → D+1
• A136572 → D+1

References

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