basis-map coefficients of A026729, shifted → basis A007318 → D+1

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Submission text ?

Triangle begins:
   1:   1;
   2:  -2,  -1;
   3:   3,   4,    1;
   4:  -4,  -9,   -6,   -1;
   5:   5,  16,   18,    8,    1;
   6:  -6, -25,  -40,  -30,  -10,   -1;
   7:   7,  36,   75,   80,   45,   12,    1;
   8:  -8, -49, -126, -175, -140,  -63,  -14,   -1;
   9:   9,  64,  196,  336,  350,  224,   84,   16,   1;
  10: -10, -81, -288, -588, -756, -630, -336, -108, -18, -1;

The row polynomials satisfy (n-1+x)*P(n,x) = -(n+(n+1)*x+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 + (n + 1)*x + x^2)*P[n - 1, x])/(n - 1 + 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 + 1)*x + x^2)*v[i - 1])/(n - 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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) 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 ?

• A131431 → x ↦ x+1 → θ+1
• A085880 → θ+1
• A178821 → basis A013609 → θ+1
• A093375 → D → D
• A085881 → basis A013609 → θ+1
• A196347 → θ+1
• A085881 → θ+1
• A038208 → D → D+1
• A196347 → basis A013609 → D+1
• A216973 → θ+1
• A344565 → θ+1
• A156992 → basis A013609 → θ+1
• A027465 → basis A013611 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A003506 → D → θ+1
• A038338 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A038338 → basis A013609 → D+1
• A135065 → θ+1
• A085880 → D → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A038330 → basis A013611 → θ+1
• A196347 → D+1
• A085881 → D+1
• A038325 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A178820 → basis A013609 → θ+1
• A178822 → basis A013609 → D+1
• A178820 → basis A013609 → D+1
• A356546 → basis A013609 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A038338 → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• basis-map coefficients of A127647 → D+1
• A356546 → θ+1
• A178822 → basis A013609 → θ+1
• A038330 → basis A013611 → D+1
• A356546 → basis A013609 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A085880 → basis A013609 → D+1
• A028326 → D → θ+1
• A038338 → θ+1
• A038208 → θ+1
• A134346 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A156992 → θ+1
• A135838 → D+1
• A134346 → D+1
• A344565 → D+1
• A028326 → D → D+1
• A178820 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A124860 → θ+1
• A156992 → basis A013609 → D+1
• A038325 → basis A013609 → D+1
• A028326 → basis A013609 → D+1
• A038221 → basis A013609 → D+1
• A028326 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A038325 → D+1
• A131431 → x ↦ x+1 → D+1
• A038208 → D+1
• A178821 → θ+1
• A085881 → basis A013609 → D+1
• A178820 → θ+1
• A178822 → θ+1
• A120910 → basis A013610 → θ+1
• A085880 → D+1
• A156992 → D+1
• A120910 → basis A013610 → D+1
• A178821 → basis A013609 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A376827 → D+1
• A085880 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A178821 → D+1
• A038221 → basis A013609 → θ+1
• A027465 → basis A013611 → θ+1
• A038221 → D+1
• A135838 → θ+1
• A038221 → θ+1
• A135065 → D+1
• A094305 → θ+1
• A376827 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A028326 → D+1
• basis-map coefficients of A127647 → θ+1
• A038325 → basis A013609 → θ+1
• A131431 → D+1 → basis A007318
• A124860 → D+1
• A038208 → D → θ+1
• A093375 → D
• A178822 → D+1
• A094305 → D+1
• A028326 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A196347 → basis A013609 → θ+1
• A356546 → D+1
• A085880 → D → 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.