A134346 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:      1;
  1:      6,       3;
  2:     28,      28,       7;
  3:    120,     180,      90,      15;
  4:    496,     992,     744,     248,      31;
  5:   2016,    5040,    5040,    2520,     630,      63;
  6:   8128,   24384,   30480,   20320,    7620,    1524,    127;
  7:  32640,  114240,  171360,  142800,   71400,   21420,   3570,    255;
  8: 130816,  523264,  915712,  915712,  572320,  228928,  57232,   8176,   511;
  9: 523776, 2356992, 4713984, 5499648, 4124736, 2062368, 687456, 147312, 18414, 1023;

The row polynomials satisfy P(n,x) = -(8+8*x+2*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(8 + 8*x + 2*x^2)*P[n - 2, x] + (6 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 8*x + 2*x^2)*v[i - 2] + (6 + 3*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 - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by 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 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
        • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
        • ↑ Liu–Wang theorem for three-term recurrences (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows, (n-1) zeros followed by 2^n - 1.?
            • ↑ Liu–Wang theorem for three-term recurrences (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).?
    • ↑ Liu–Wang theorem for three-term recurrences (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729 → basis A013609 → D
• A178822 → x ↦ x/(x+1)
• A178820 → basis A007318
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A038221 → basis A013609 → basis A013611
• A131431 → x ↦ x+1 → basis A007318
• A028326 → basis A007318 → D
• A038338 → x ↦ x+1 → reverse
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A132440 → θ+1 → basis A013609
• A085881 → basis A013611 → reverse
• A356546 → x ↦ x+1 → D
• A356546 → x ↦ x+1
• A038208 → basis A013611 → D
• A013609 → D
• A196347 → basis A013611 → D
• A120909 → basis A013610 → reverse
• A085880 → basis A013609 → basis A013611
• A085881 → basis A007318
• A013609 → D → basis A013610
• basis-map coefficients of A026729 → D → basis A013609
• A159854 → x ↦ x/(x+1)
• A003506 → D → x ↦ x+1
• A356546 → basis A013611 → reverse
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A196347 → basis A013609 → basis A007318
• A085880 → basis A013611 → basis A013610
• A216973 → basis A013611
• A178822 → x ↦ x+1
• A134309 → basis A013609
• A129186 → basis A013609
• A178820 → basis A013611 → basis A013610
• A356546 → basis A013611 → D
• A094305 → basis A013611
• A196347 → basis A013611 → basis A013610
• A135838 → x ↦ x/(x+1)
• A038221 → basis A013611 → D
• A178820 → basis A007318 → D
• A038221 → x ↦ x+1
• A038325 → basis A007318 → reverse
• A376827 → x ↦ x/(x+1)
• A134058 → basis A007318
• basis-map coefficients of A112227 → basis A013609 → D
• basis-map coefficients of A112227 → x ↦ x+1 → basis A007318
• A130123 → θ+1 → basis A013609
• A134059 → basis A007318
• A156992 → basis A007318 → D
• A130123 → basis A013609
• A027465 → basis A013611 → basis A013611
• A132200 → basis A007318
• A136572 → (θ+1)² → basis A013609
• A178821 → basis A013611
• A038330 → basis A013609 → reverse
• A038338 → basis A013611
• A135089 → basis A013611
• A344565 → basis A013611
• basis-map coefficients of A127647 → x ↦ x/(x+1)
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A131431 → D → basis A013609
• A356546 → basis A013611
• A038208 → basis A013611
• A120909 → basis A013610 → basis A013610
• A196347 → x ↦ x+1
• A120910 → basis A013610 → basis A013611
• A178821 → basis A007318 → reverse
• A344565 → basis A007318
• A196347 → basis A013611
• A134058 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013610
• A156992 → basis A013611 → basis A013610
• A136521 → basis A013609
• basis-map coefficients of A113214, shifted → D → basis A013611
• A178822 → basis A013609 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → reverse
• A003506 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A007318 → basis A013611
• A131431 → θ+1 → basis A013609
• A028326 → basis A013611 → basis A013610
• Veronese parent of A098172 → x ↦ x/(x+1)
• A120910 → D → D
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → x ↦ x+1
• A038330 → basis A013611 → basis A007318
• A376827 → basis A007318
• A258758 → basis A013611
• A085881 → x ↦ x/(x+1)
• A027466 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A028326 → basis A013609 → x ↦ x+1
• basis-map coefficients of A113214, shifted → basis A013611
• A085881 → basis A013611 → D
• basis-map coefficients of A026729, shifted → basis A013609 → D
• A135065 → x ↦ x+1 → reverse
• A131431 → x ↦ x+1 → x ↦ x/(x+1)
• A156992 → x ↦ x/(x+1)
• A120909 → basis A013610
• A120909 → D
• A038291 → basis A013609 → basis A013610
• A178822 → basis A013611 → basis A013610
• A013609 → basis A013610
• basis-map coefficients of A030528, shifted → D → basis A013609
• A356546 → x ↦ x+1 → reverse
• A135065 → x ↦ x+1 → D
• image under θ+1 → basis A013609
• A085880 → D → basis A007318
• A132200 → basis A013611
• A147716 → basis A013610 → basis A013610
• A178820 → x ↦ x/(x+1)
• A258758 → basis A007318
• A038330 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → basis A013609 → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A028326 → basis A013611 → D
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A085881 → basis A007318 → reverse
• A038325 → basis A007318 → D
• A178822 → basis A013611
• basis-map coefficients of A113214, shifted → basis A013609 → basis A013611
• A013609 → basis A013610 → basis A013610
• A178821 → basis A007318 → D
• A085880 → x ↦ x+1 → reverse
• A127648 → D → basis A013609
• image under θ+1 → θ+1 → D → basis A013609
• A038325 → basis A013611
• A038221 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609
• basis-map coefficients of A127647 → basis A013611
• A003506 → D → basis A013611
• A085881 → basis A007318 → D
• Veronese parent of A098172 → x ↦ x/(x+1)
• A094305 → x ↦ x/(x+1)
• A124860 → x ↦ x+1
• A028326 → basis A007318
• A013609 → D → D
• A156992 → basis A007318
• A135065 → x ↦ x+1
• A136572 → θ+1 → basis A013609
• A178821 → basis A013609 → basis A013611
• A178821 → basis A013611 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013609
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013611
• A356546 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013610
• A038221 → x ↦ x+1 → D
• A178822 → basis A013611 → D
• A130123 → D → basis A013609
• A178822 → basis A013609 → x ↦ x+1
• basis-map coefficients of A112227 → basis A013609 → basis A013610
• A038329 → x ↦ x+1 → basis A013610
• A131431 → basis A013609
• A178821 → basis A013611 → basis A013610
• A028326 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• basis-map coefficients of A112227 → θ+1 → basis A013609
• A003506 → x ↦ x+1
• A085880 → basis A013609 → x ↦ x+1
• A085881 → basis A013611 → basis A013610
• basis-map coefficients of A026729, shifted → D → basis A013609
• A178821 → basis A007318
• A038338 → x ↦ x+1 → D
• A178821 → basis A013611 → reverse
• A156992 → basis A013609 → basis A007318
• A178820 → basis A013611 → reverse
• A028326 → x ↦ x/(x+1)
• A124860 → basis A013611
• A085880 → basis A013611 → D
• A178820 → basis A007318 → reverse
• basis-map coefficients of A026729 → basis A013609
• A178822 → x ↦ x+1 → D
• basis-map coefficients of A030528, shifted → θ+1 → basis A013609
• A120910 → D
• A178821 → x ↦ x/(x+1)
• A124860 → x ↦ x/(x+1)
• A085880 → x ↦ x+1
• A013609 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013609 → D
• A120910 → basis A007318 → basis A013609
• A094305 → x ↦ x+1 → reverse
• A085880 → x ↦ x/(x+1)
• A038330 → basis A013609
• A136572 → D → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609
• A134346 → x ↦ x/(x+1)
• A130123 → basis A013609 → D
• A135065 → x ↦ x/(x+1)
• basis-map coefficients of A127647 → basis A007318
• A135065 → basis A013611
• basis-map coefficients of A026729, shifted → θ+1 → basis A013609
• A356546 → basis A013611 → basis A013610
• basis-map coefficients of A021012 → θ+1 → basis A013609
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013609
• A196347 → x ↦ x+1 → D
• A178822 → x ↦ x+1 → reverse
• A038208 → D → basis A013611
• A085880 → basis A013611
• A038338 → x ↦ x/(x+1)
• A038325 → basis A013609 → basis A007318
• basis-map coefficients of A026729 → (θ+1)² → basis A013609
• A038325 → x ↦ x/(x+1)
• A136572 → basis A013609 → D
• A028326 → basis A007318 → reverse
• A376827 → basis A013611
• A094305 → x ↦ x+1
• A038208 → x ↦ x/(x+1)
• A038330 → basis A013609 → D
• A038221 → basis A013611 → basis A013610
• A028326 → basis A013609 → basis A013611
• A156992 → basis A013611
• A344565 → x ↦ x/(x+1)
• A356546 → basis A013609 → basis A013611
• A135838 → basis A013611
• A120909 → basis A013610 → D
• A094305 → x ↦ x+1 → D
• image under θ+1 → (θ+1)² → basis A013609
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013611
• A027465 → D → basis A013609
• A038338 → basis A013609 → basis A007318
• A178820 → basis A013611
• A131431 → x ↦ x+1 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013610
• A132440 → basis A013609
• A134346 → basis A013611
• A038325 → basis A007318
• basis-map coefficients of A112227 → basis A013609
• A131436 → basis A013609
• A178820 → basis A013611 → D
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A135838 → x ↦ x+1
• A178820 → basis A013609 → basis A013611
• A085881 → basis A013611
• A136572 → basis A013609 → basis A013610
• A135089 → basis A007318
• A156992 → basis A013611 → D
• A038208 → basis A007318 → D
• basis-map coefficients of A113214, shifted → D → basis A007318
• A027466 → basis A013611 → D
• A134059 → basis A013611
• A038338 → x ↦ x+1
• A038208 → basis A007318
• Veronese parent of A098172 → x ↦ x/(x+1)
• A085880 → D → basis A013611
• A136572 → basis A013609

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003