A120909 → basis A013610 → θ+1

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Submission text ?

Triangle begins:
  0:     3;
  1:     9,      36;
  2:    27,     216,      324;
  3:    81,     972,     2916,      2592;
  4:   243,    3888,    17496,     31104,     19440;
  5:   729,   14580,    87480,    233280,    291600,     139968;
  6:  2187,   52488,   393660,   1399680,   2624400,    2519424,     979776;
  7:  6561,  183708,  1653372,   7348320,  18370800,   26453952,   20575296,    6718464;
  8: 19683,  629856,  6613488,  35271936, 110224800,  211631616,  246903552,  161243136,   45349632;
  9: 59049, 2125764, 25509168, 158723712, 595213920, 1428513408, 2222131968, 2176782336, 1224440064, 302330880;

The row polynomials satisfy (1+2*n*x)*P(n,x) = (3+(6*n+12)*x+(12*n+12)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.

P[0, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((3 + (6*n + 12)*x + (12*n + 12)*x^2)*P[n - 1, x])/(1 + 2*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((3 + (6*n + 12)*x + (12*n + 12)*x^2)*v[i - 1])/(1 + 2*n*x)); 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 (−∞, 0]
  by Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subw…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…?
        • 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
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • 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 Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal lengt…?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
                • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, …?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A lev…?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ 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 basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, …?
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A156992 → basis A013611 → θ+1
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
• A028326 → basis A013611 → θ+1
• A120909 → θ+1
• A196347 → basis A013611 → θ+1
• A178820 → basis A013611 → θ+1
• A013609 → basis A013610 → θ+1
• A356546 → basis A013611 → θ+1
• A013609 → D → θ+1
• A178822 → basis A013611 → θ+1
• A038208 → basis A013611 → θ+1
• A136572 → basis A013609 → θ+1
• A116666 → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1
• A085880 → basis A013611 → θ+1
• A085881 → basis A013611 → θ+1
• A038221 → basis A013611 → θ+1
• A130123 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A116666 → reverse → x ↦ x+1
• A134082 → basis A007318 → x ↦ x+1
• A013609 → θ+1
• basis-map coefficients of A112227 → basis A013609 → θ+1
• A178821 → basis A013611 → θ+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. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.