basis-map coefficients of A021012 → basis A008277 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
   1:       1;
   2:      -1,        -1;
   3:       2,         6,          2;
   4:      -6,       -36,        -42,          -6;
   5:      24,       240,        600,         360,          24;
   6:    -120,     -1800,      -7800,      -10800,       -3720,         -120;
   7:     720,     15120,     100800,      252000,      216720,        45360,          720;
   8:   -5040,   -141120,   -1340640,    -5292000,    -8573040,     -4868640,      -640080,       -5040;
   9:   40320,   1451520,   18627840,   106686720,   280264320,    313286400,    121968000,    10281600,      40320;
  10: -362880, -16329600, -272160000, -2133734400, -8283461760, -15431472000, -12376022400, -3385670400, -185431680, -362880;

The row polynomials satisfy P(n,x) = -(n-1+(n^2-2*n+1)*x)*P(n-1,x) + (n-1)*x^2*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 - 1 + (n^2 - 2*n + 1)*x)*P[n - 1, x] + (n - 1)*x^2*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 - 1 + (n^2 - 2*n + 1)*x)*v[i - 1] + (n - 1)*x^2*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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A008277 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polyno…?
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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)
    • Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? 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)

Related ?

Same fingerprint ?

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

References

1. [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