basis-map coefficients of A026729, shifted → basis A007318 → basis A008277

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Submission text ?

Triangle begins:
   1:    1;
   2:   -2,     -1;
   3:    4,      5,       1;
   4:   -8,    -19,      -9,      -1;
   5:   16,     65,      55,      14,       1;
   6:  -32,   -211,    -285,    -125,     -20,     -1;
   7:   64,    665,    1351,     910,     245,     27,      1;
   8: -128,  -2059,   -6069,   -5901,   -2380,   -434,    -35,    -1;
   9:  256,   6305,   26335,   35574,   20181,   5418,    714,    44,   1;
  10: -512, -19171, -111645, -204205, -156660, -58107, -11130, -1110, -54, -1;

The row polynomials satisfy P(n,x) = -(2+x)*P(n-1,x) - 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[-(2 + x)*P[n - 1, x] - 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] = -(2 + x)*v[i - 1] - 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 shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
            • ↑ Parent-row provers of polytool (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >…?
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A376827 → basis A008277
• A178820 → basis A013609 → basis A008277
• A094305 → basis A008277
• A085881 → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A143494 → reverse
• A178821 → basis A013609 → basis A008277
• A135065 → basis A008277
• A038221 → basis A008277
• A038338 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A344565 → basis A008277
• A178822 → basis A013609 → basis A008277
• A085880 → D → basis A008277
• A356546 → basis A013609 → basis A008277
• A356546 → basis A008277
• A178822 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A156992 → basis A008277
• A135089 → basis A008277
• A028326 → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A038330 → basis A013611 → basis A008277
• A038208 → D → basis A008277
• A003506 → basis A008277
• A124860 → basis A008277
• A038208 → basis A008277
• A196347 → basis A008277
• A028326 → D → basis A008277
• A156992 → basis A013609 → basis A008277
• A003506 → D → basis A008277
• A085880 → basis A008277
• basis-map coefficients of A113214, shifted → D → basis A008277
• A038221 → basis A013609 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A132200 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A085881 → basis A013609 → basis A008277
• A135838 → basis A008277
• A196347 → basis A013609 → basis A008277
• basis-map coefficients of A021012 → basis A008277 → D+1
• A028326 → basis A013609 → basis A008277
• A038325 → basis A013609 → basis A008277
• A027465 → basis A013611 → basis A008277
• A085880 → basis A013609 → basis A008277
• A156992 → basis A008277 → reverse
• A126351 → reverse
• A178821 → basis A008277
• A038325 → basis A008277
• A134058 → basis A008277
• A134059 → basis A008277
• A134346 → basis A008277
• A120910 → basis A013610 → basis A008277
• A038338 → basis A013609 → basis A008277
• A258758 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A178820 → 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