basis-map coefficients of A112227 → basis A013611 → x ↦ x/(x+1)

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Triangle begins:
   1: 1, 10,   25;
   2: 1, 15,   75,   125;
   3: 1, 20,  150,   500,    625;
   4: 1, 25,  250,  1250,   3125,    3125;
   5: 1, 30,  375,  2500,   9375,   18750,   15625;
   6: 1, 35,  525,  4375,  21875,   65625,  109375,    78125;
   7: 1, 40,  700,  7000,  43750,  175000,  437500,   625000,   390625;
   8: 1, 45,  900, 10500,  78750,  393750, 1312500,  2812500,  3515625,   1953125;
   9: 1, 50, 1125, 15000, 131250,  787500, 3281250,  9375000, 17578125,  19531250,   9765625;
  10: 1, 55, 1375, 20625, 206250, 1443750, 7218750, 25781250, 64453125, 107421875, 107421875, 48828125;

The row polynomials satisfy (1+x)*P(n,x) = (1+6*x+5*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 10*x + 25*x^2.

P[1, x_] := 1 + 10*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 6*x + 5*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 10*x + 25*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + 6*x + 5*x^2)*v[i - 1])/(1 + x)); 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, 0]
  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 Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis-map coefficients of A scaled Hermite triangle.?
        • 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 A scaled Hermite triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 of basis-map coefficients of A scaled Hermite triangle.?
        • Geometric basis map (elementary) polytool ? step ›

          • from basis-map coefficients of A scaled Hermite triangle.?
            • ↑ 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 basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.?
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of A scaled Hermite triangle.?
                • ↑ 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 shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from basis-map coefficients of A scaled Hermite triangle.?
                    • ↑ 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 shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.?
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                    • Geometric basis map (elementary) polytool ? step ›

                      • from basis-map coefficients of A scaled Hermite triangle.?
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013612 → D
• Triangle of coefficients in expansion of (1+5x)^n.
• A038329 → x ↦ x+1 → reverse
• A136572 → basis A013611 → x ↦ x/(x+1)
• A038291 → basis A013609 → reverse
• A027467 → basis A013611 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013611 → x ↦ x/(x+1)
• A013612 → reverse
• A038243 → D
• A013611 → x ↦ x/(x+1)
• A038329 → x ↦ x+1
• A038291 → basis A013609
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
• A038291 → basis A013609 → D
• A147716 → basis A013610
• A038329 → x ↦ x+1 → D
• A038330 → basis A007318 → basis A007318
• A130123 → basis A013611 → x ↦ x/(x+1)
• A038243 → reverse
• A147716 → basis A013610 → reverse
• A147716 → basis A013610 → D

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.