basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²

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Triangle begins:
  0:    1;
  1:   -2,     -8;
  2:    4,     32,      36;
  3:   -8,    -96,    -216,    -128;
  4:   16,    256,     864,    1024,      400;
  5:  -32,   -640,   -2880,   -5120,    -4000,    -1152;
  6:   64,   1536,    8640,   20480,    24000,    13824,     3136;
  7: -128,  -3584,  -24192,  -71680,  -112000,   -96768,   -43904,    -8192;
  8:  256,   8192,   64512,  229376,   448000,   516096,   351232,   131072,   20736;
  9: -512, -18432, -165888, -688128, -1612800, -2322432, -2107392, -1179648, -373248, -51200;

The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = -(2+(6*n+6)*x+(2*n^2+10*n+6)*x^2+(2*n^2+4*n+2)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2 + (6*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^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] = (-(2 + (6*n + 6)*x + (2*n^2 + 10*n + 6)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*x^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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
            • Geometric basis map (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 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)
            • Parent-row provers of polytool 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) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0?
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A003506 → D → (θ+1)²
• A085880 → (θ+1)²
• A125092 → reverse
• A344565 → (θ+1)²
• A038208 → D → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A038325 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• A038208 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• A156992 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A038338 → basis A013609 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A038221 → (θ+1)²
• A003506 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A085881 → (θ+1)²
• A085880 → D → (θ+1)²
• A356546 → (θ+1)²
• A134346 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A178820 → (θ+1)²
• A376827 → (θ+1)²
• A178821 → (θ+1)²
• A120910 → basis A013610 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• A125092 → reverse → D
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A038338 → (θ+1)²
• A135838 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A178822 → (θ+1)²
• A028326 → (θ+1)²
• A028326 → D → (θ+1)²
• A135065 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A124860 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A196347 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²

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.