basis-map coefficients of A026729 → basis A013610 → D

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Triangle begins:
   1:  0;
   2:  3;
   3:  6,  18;
   4:  9,  54,   81;
   5: 12, 108,  324,   324;
   6: 15, 180,  810,  1620,   1215;
   7: 18, 270, 1620,  4860,   7290,   4374;
   8: 21, 378, 2835, 11340,  25515,  30618,  15309;
   9: 24, 504, 4536, 22680,  68040, 122472, 122472,  52488;
  10: 27, 648, 6804, 40824, 153090, 367416, 551124, 472392, 177147;

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

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 3)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(n - 2)); 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 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…?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • 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…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (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) 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…?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ 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) 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…?
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ 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) 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…?
                    • ↑ Iterated real-rootedness preservers (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
                    • 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
                    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
                    • 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
                    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).?
                    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
                    • Parent-row provers of polytool polytool ? step › direct check
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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

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.