basis-map coefficients of A030528, shifted → (θ+1)² → basis A013610

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Triangle begins:
  0:    1;
  1:   -4,   -12;
  2:    9,    54,     81;
  3:  -16,  -144,   -432,    -432;
  4:   25,   300,   1350,    2700,     2025;
  5:  -36,  -540,  -3240,   -9720,   -14580,    -8748;
  6:   49,   882,   6615,   26460,    59535,    71442,    35721;
  7:  -64, -1344, -12096,  -60480,  -181440,  -326592,  -326592,  -139968;
  8:   81,  1944,  20412,  122472,   459270,  1102248,  1653372,  1417176,   531441;
  9: -100, -2700, -32400, -226800, -1020600, -3061800, -6123600, -7873200, -5904900, -1968300;

The row polynomials satisfy n^2*P(n,x) = -(n^2+2*n+1+(3*n^2+6*n+3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 2*n + 1 + (3*n^2 + 6*n + 3)*x)*P[n - 1, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 2*n + 1 + (3*n^2 + 6*n + 3)*x)*v[i - 1])/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from theta_plus_one of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x…?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

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

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.