basis-map coefficients of A026729, shifted → basis A013609 → basis A007318

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Triangle begins:
   1:      1;
   2:     -3,      -2;
   3:      9,      12,       4;
   4:    -27,     -54,     -36,      -8;
   5:     81,     216,     216,      96,      16;
   6:   -243,    -810,   -1080,    -720,    -240,     -32;
   7:    729,    2916,    4860,    4320,    2160,     576,      64;
   8:  -2187,  -10206,  -20412,  -22680,  -15120,   -6048,   -1344,   -128;
   9:   6561,   34992,   81648,  108864,   90720,   48384,   16128,   3072,   256;
  10: -19683, -118098, -314928, -489888, -489888, -326592, -145152, -41472, -6912, -512;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(3 + 2*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] = -(3 + 2*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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
    • 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 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 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 (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (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)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013609 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k …?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A028326 → basis A013610 → D
• A038338 → basis A013610
• A120910 → x ↦ x/(x+1)
• A085880 → basis A013610
• A178820 → basis A007318 → basis A013609
• A135065 → x ↦ x+1 → basis A013609
• A038328 → basis A013609 → basis A013610
• A120909 → basis A013610 → x ↦ x+1
• A196347 → basis A013610
• A027466 → basis A013611 → x ↦ x/(x+1)
• A156992 → basis A007318 → basis A013609
• A344565 → basis A013610
• A178821 → basis A013611 → basis A007318
• A013610 → basis A013611 → basis A013609
• A196347 → basis A013610 → D
• A196347 → basis A013611 → x ↦ x+1
• A178821 → basis A013610 → reverse
• A178822 → x ↦ x+1 → basis A013609
• basis-map coefficients of A113214, shifted → D → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A028326 → basis A013610
• A038221 → basis A013610 → D
• A038208 → D → basis A013610
• A124860 → basis A013610
• A135089 → basis A013610
• A134059 → basis A013610
• A120910 → basis A013609 → D
• A085881 → basis A007318 → basis A013609
• A178822 → basis A013610
• A178820 → basis A013609 → basis A013610
• A038221 → basis A013611 → x ↦ x+1
• A356546 → basis A013609 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A134058 → basis A013610
• A013620 → D
• A120910 → basis A013609 → reverse
• A156992 → basis A013609 → basis A013610
• A038338 → basis A013609 → basis A013610
• A038338 → basis A013610 → reverse
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A038221 → basis A013609 → basis A013610
• A216973 → basis A013610
• A178820 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A258758 → basis A013610
• A120910 → basis A013610 → basis A013610
• A356546 → basis A013611 → basis A007318
• A013610 → D → basis A013609
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A013609 → basis A013610 → x ↦ x+1
• A356546 → basis A013610
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A038220 → D
• A038330 → basis A013609 → basis A013609
• A135838 → basis A013610
• A356546 → x ↦ x+1 → basis A013609
• A135065 → basis A013610
• A156992 → basis A013610
• A038208 → basis A013610
• A038338 → basis A013610 → D
• A038325 → basis A013610
• A038325 → basis A007318 → basis A013609
• A120910 → basis A013609
• A094305 → x ↦ x+1 → basis A013609
• basis-map coefficients of A113214, shifted → basis A013610
• A038221 → basis A013610
• A038243 → basis A013611
• A136572 → basis A013610 → basis A013609
• A028326 → basis A013611 → x ↦ x+1
• A003506 → D → basis A013610
• A085880 → D → basis A013610
• A038291 → basis A013609 → basis A013611
• A178821 → basis A013610 → D
• A038208 → basis A013610 → D
• A038334 → reverse
• A027465 → basis A013611 → basis A013610
• A085881 → basis A013611 → basis A007318
• A085881 → basis A013610
• A028326 → basis A013609 → basis A013610
• A085880 → basis A013609 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A120910 → D → basis A013609
• A178820 → basis A013611 → basis A007318
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A085880 → basis A013610 → reverse
• A028326 → D → basis A013610
• A178822 → basis A013609 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A132200 → basis A013610
• A156992 → basis A013611 → x ↦ x+1
• A131431 → x ↦ x+1 → basis A013610
• A085880 → basis A013611 → x ↦ x+1
• A156992 → basis A013610 → D
• A038330 → basis A013609 → x ↦ x/(x+1)
• A085880 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A038243 → basis A013611 → D
• basis-map coefficients of A127647 → basis A013610
• A038334 → D
• A178820 → basis A013610 → D
• A038325 → basis A013609 → basis A013610
• A028326 → basis A007318 → basis A013609
• A178821 → basis A013610
• A136572 → basis A013609 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A134346 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A094305 → basis A013610
• A085881 → basis A013609 → basis A013610
• A003506 → x ↦ x+1 → basis A013609
• A038330 → basis A013611 → basis A013610
• basis-map coefficients of A026729 → basis A013609 → x ↦ x+1
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A376827 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A178822 → basis A013610 → D
• A178822 → basis A013611 → basis A007318
• A178821 → basis A013609 → basis A013610