basis-map coefficients of A112227 → basis A013610 → basis A013611

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Submission text ?

Triangle begins:
   1:      16,        96,        144;
   2:      64,       576,       1728,        1728;
   3:     256,      3072,      13824,       27648,        20736;
   4:    1024,     15360,      92160,      276480,       414720,       248832;
   5:    4096,     73728,     552960,     2211840,      4976640,      5971968,       2985984;
   6:   16384,    344064,    3096576,    15482880,     46448640,     83607552,      83607552,      35831808;
   7:   65536,   1572864,   16515072,    99090432,    371589120,    891813888,    1337720832,    1146617856,     429981696;
   8:  262144,   7077888,   84934656,   594542592,   2675441664,   8026324992,   16052649984,   20639121408,   15479341056,    5159780352;
   9: 1048576,  31457280,  424673280,  3397386240,  17836277760,  64210599936,  160526499840,  275188285440,  309586821120,  206391214080,   61917364224;
  10: 4194304, 138412032, 2076180480, 18685624320, 112113745920, 470877732864, 1412633198592, 3027071139840, 4540606709760, 4540606709760, 2724364025856, 743008370688;

The row polynomials satisfy P(n,x) = (4+12*x)*P(n-1,x) for n >= 2, with P(1,x) = 16 + 96*x + 144*x^2.

P[1, x_] := 16 + 96*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 16 + 96*x + 144*x^2; for(n = 2, N, my(i = n); v[i] = (4 + 12*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 (−∞, 0]
  by Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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 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_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 ?

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