basis-map coefficients of A021012 → θ+1 → basis A008277

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:        1;
   2:       -2,          -2;
   3:        6,          18,            6;
   4:      -24,        -168,         -144,           -24;
   5:      120,        1800,         3000,          1200,           120;
   6:     -720,      -22320,       -64800,        -46800,        -10800,         -720;
   7:     5040,      317520,      1517040,       1764000,        705600,       105840,         5040;
   8:   -40320,    -5120640,    -38949120,     -68584320,     -42336000,    -10725120,     -1128960,      -40320;
   9:   362880,    92534400,   1097712000,    2819577600,    2522378880,    960180480,    167650560,    13063680,     362880;
  10: -3628800, -1854316800, -33856704000, -123760224000, -154314720000, -82834617600, -21337344000, -2721600000, -163296000, -3628800;

The row polynomials satisfy P(n,x) = -(n+n*x)*P(n-1,x) - n*x*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[-(n + n*x)*P[n - 1, x] - n*x*D[P[n - 1, x], 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 + n*x)*v[i - 1] - n*x*deriv(v[i - 1], 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)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomia…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
            • ↑ Parent-row provers of polytool (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
                • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomia…?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A112227 → θ+1 → basis A008277
• A132440 → θ+1 → basis A008277
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
• A136521 → basis A008277
• image under θ+1 → basis A008277
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
• basis-map coefficients of A026729 → D → basis A008277
• A136572 → θ+1 → basis A008277
• A048993 → D+1
• A131431 → θ+1 → basis A008277
• A130123 → D → basis A008277
• A136572 → D → basis A008277
• A136572 → basis A008277
• basis-map coefficients of A026729 → (θ+1)² → basis A008277
• A048993 → D+1 → reverse
• basis-map coefficients of A021012 → basis A008277 → reverse
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A008277
• A134309 → basis A008277
• A127648 → D → basis A008277
• basis-map coefficients of A026729, shifted → D → basis A008277
• A131431 → basis A008277
• image under θ+1 → (θ+1)² → basis A008277
• basis-map coefficients of A030528, shifted → D → basis A008277
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A008277
• basis-map coefficients of A026729, shifted → basis A008277
• basis-map coefficients of A026729, shifted → θ+1 → basis A008277
• A129186 → basis A008277
• A130123 → θ+1 → basis A008277
• A131436 → basis A008277
• basis-map coefficients of A112227 → basis A008277 → reverse
• A131431 → D → basis A008277
• basis-map coefficients of A030528, shifted → basis A008277
• A136572 → (θ+1)² → basis A008277
• A048993 → reverse
• A130123 → basis A008277 → reverse
• image under θ+1 → θ+1 → D → basis A008277
• basis-map coefficients of A112227 → (θ+1)² → basis A008277
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
• basis-map coefficients of A112227 → basis A008277
• basis-map coefficients of A030528, shifted → θ+1 → basis A008277
• basis-map coefficients of A021012 → basis A008277
• A130123 → basis A008277
• A132440 → basis A008277
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
• A130123 → (θ+1)² → basis A008277

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.