basis-map coefficients of A026729, shifted → θ+1 → basis A013610

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Triangle begins:
   1:   1;
   2:  -2,   -6;
   3:   3,   18,    27;
   4:  -4,  -36,  -108,   -108;
   5:   5,   60,   270,    540,     405;
   6:  -6,  -90,  -540,  -1620,   -2430,   -1458;
   7:   7,  126,   945,   3780,    8505,   10206,    5103;
   8:  -8, -168, -1512,  -7560,  -22680,  -40824,  -40824,  -17496;
   9:   9,  216,  2268,  13608,   51030,  122472,  183708,  157464,   59049;
  10: -10, -270, -3240, -22680, -102060, -306180, -612360, -787320, -590490, -196830;

The row polynomials satisfy (n-1)*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[Together[(-(n + 3*n*x)*P[n - 1, x])/(n - 1)]];
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])/(n - 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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= …?
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

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

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.