basis-map coefficients of A021012 → θ+1 → basis A013610

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Triangle begins:
   1:        1;
   2:       -2,        -6;
   3:        6,        36,          54;
   4:      -24,      -216,        -648,        -648;
   5:      120,      1440,        6480,       12960,         9720;
   6:     -720,    -10800,      -64800,     -194400,      -291600,       -174960;
   7:     5040,     90720,      680400,     2721600,      6123600,       7348320,       3674160;
   8:   -40320,   -846720,    -7620480,   -38102400,   -114307200,    -205752960,    -205752960,     -88179840;
   9:   362880,   8709120,    91445760,   548674560,   2057529600,    4938071040,    7407106560,    6348948480,    2380855680;
  10: -3628800, -97977600, -1175731200, -8230118400, -37035532800, -111106598400, -222213196800, -285702681600, -214277011200, -71425670400;

The row polynomials satisfy P(n,x) = -(n+3*n*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n + 3*n*x)*P[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 + 3*n*x)*v[i - 1]); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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 [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 A030528, shifted → (θ+1)² → basis A013610
• A136572 → θ+1 → basis A013610
• A136572 → basis A013610 → basis A013611
• A136572 → (θ+1)² → basis A013610
• A038330 → D
• A178821 → basis A007318 → x ↦ x+1
• basis-map coefficients of A112227 → basis A013610 → D
• basis-map coefficients of A026729 → D → basis A013610
• basis-map coefficients of A112227 → basis A013610
• A196347 → basis A013611 → x ↦ x/(x+1)
• A135065 → x ↦ x+1 → basis A007318
• basis-map coefficients of A112227 → basis A013610 → basis A013611
• A085880 → x ↦ x+1 → x ↦ x+1
• basis-map coefficients of A026729, shifted → D → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → D
• A013610 → basis A013611 → D
• basis-map coefficients of A026729, shifted → basis A013610
• A038243 → basis A013609
• A136572 → basis A013610
• A147716 → basis A013610 → basis A013609
• A038221 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → D → basis A013610
• A120909 → x ↦ x/(x+1)
• A027465: Cube of lower triangular normalized binomial matrix.
• A156992 → basis A007318 → x ↦ x+1
• A129186 → basis A013610
• A027465 → D
• A130123 → (θ+1)² → basis A013610
• A132440 → θ+1 → basis A013610
• A038325 → basis A007318 → basis A007318
• basis-map coefficients of A026729 → basis A013610
• A178821 → basis A013611 → x ↦ x/(x+1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A131436 → basis A013610
• A028326 → basis A007318 → basis A007318
• basis-map coefficients of A026729 → basis A013610 → reverse
• A120910 → basis A007318
• basis-map coefficients of A026729, shifted → θ+1 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → D
• basis-map coefficients of A112227 → basis A013610 → reverse
• A130123 → basis A013610 → basis A013611
• A136572 → basis A013610 → D
• A013610 → D → D
• basis-map coefficients of A026729, shifted → basis A013610 → reverse
• A038330 → basis A013609 → x ↦ x+1
• A127648 → D → basis A013610
• A120910 → D → x ↦ x+1
• basis-map coefficients of A112227 → (θ+1)² → basis A013610
• A134309 → basis A013610
• A013610 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → reverse
• A094305 → x ↦ x+1 → x ↦ x+1
• A136521 → basis A013610
• A127648 → basis A013610
• A136572 → D → basis A013610
• basis-map coefficients of A112227 → θ+1 → basis A013610
• A038243 → basis A013609 → D
• basis-map coefficients of A030528, shifted → basis A013610
• A085880 → basis A013611 → x ↦ x/(x+1)
• A130123 → D → basis A013610
• basis-map coefficients of A026729 → basis A013610 → D
• basis-map coefficients of A030528, shifted → θ+1 → basis A013610
• A130123 → basis A013610
• A178820 → basis A007318 → basis A007318
• A085881 → basis A013611 → x ↦ x/(x+1)
• A038338 → x ↦ x+1 → basis A007318
• A038329 → x ↦ x+1 → basis A013609
• image under θ+1 → θ+1 → D → basis A013610
• A120910 → basis A007318 → reverse
• A131431 → basis A013610
• A356546 → x ↦ x+1 → x ↦ x+1
• A085881 → basis A007318 → basis A007318
• A038279 → basis A013610 → D
• basis-map coefficients of A026729 → basis A013610 → basis A013611
• A178822 → x ↦ x+1 → x ↦ x+1
• A013609 → basis A013610 → x ↦ x/(x+1)
• A196347 → x ↦ x+1 → basis A007318
• A027465 → D → D
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013611
• A120909 → basis A013610 → x ↦ x/(x+1)
• image under θ+1 → basis A013610
• A131431 → D → basis A013610
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013610
• A130123 → θ+1 → basis A013610
• A156992 → basis A013611 → x ↦ x/(x+1)
• A356546 → basis A013611 → x ↦ x/(x+1)
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A178820 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → (θ+1)² → basis A013610
• A131431 → θ+1 → basis A013610
• A028326 → basis A013611 → x ↦ x/(x+1)
• A130123 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013611
• basis-map coefficients of A021012 → basis A013610
• A178822 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013611 → reverse
• A038330 → reverse
• A038279 → basis A013610
• A013610 → D → basis A013611
• A013610 → D
• image under θ+1 → (θ+1)² → basis A013610
• A038279 → basis A013610 → reverse

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