basis-map coefficients of A026729 → basis A013609 → x ↦ x+1

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Triangle begins:
   1:     1;
   2:     3,      2;
   3:     9,     12,      4;
   4:    27,     54,     36,      8;
   5:    81,    216,    216,     96,     16;
   6:   243,    810,   1080,    720,    240,     32;
   7:   729,   2916,   4860,   4320,   2160,    576,     64;
   8:  2187,  10206,  20412,  22680,  15120,   6048,   1344,   128;
   9:  6561,  34992,  81648, 108864,  90720,  48384,  16128,  3072,  256;
  10: 19683, 118098, 314928, 489888, 489888, 326592, 145152, 41472, 6912, 512;

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

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

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (3 + 2*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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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 [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 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)
• 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) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A003506 → D → basis A013610
• A356546 → basis A013609 → basis A013610
• A028326 → D → basis A013610
• A038221 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013609 → x ↦ x+1
• A038330 → basis A013611 → basis A013610
• basis-map coefficients of A112227 → basis A013610 → basis A013609
• A085880 → basis A013610 → reverse
• A038243 → basis A013611 → D
• A134059 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A085881 → basis A013610
• A136572 → basis A013610 → basis A013609
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
• A038208 → basis A013610 → D
• A038338 → basis A013610
• A013610 → D → basis A013609
• A038325 → basis A013609 → basis A013610
• A196347 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A038338 → basis A013609 → basis A013610
• basis-map coefficients of A026729, shifted → basis A007318 → basis A013610
• A038291 → basis A013609 → basis A013611
• A124860 → basis A013610
• basis-map coefficients of A026729 → basis A007318 → basis A013610
• A135065 → basis A013610
• A258758 → basis A013610
• A028326 → basis A013609 → basis A013610
• A085881 → basis A013609 → basis A013610
• A120910 → x ↦ x/(x+1)
• A038338 → basis A013610 → reverse
• A085881 → basis A007318 → basis A013609
• A038325 → basis A007318 → basis A013609
• A094305 → x ↦ x+1 → x ↦ x/(x+1)
• A120910 → D → basis A013609
• A038220 → D
• A178821 → basis A013610
• A120910 → basis A013610 → basis A013610
• A013620 → D
• A135838 → basis A013610
• A038334 → reverse
• A038221 → basis A013611 → x ↦ x+1
• A178821 → basis A013609 → basis A013610
• A156992 → basis A013610
• A038221 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013609
• A178822 → basis A013610 → D
• A028326 → basis A013610 → D
• A344565 → basis A013610
• A178820 → basis A013611 → basis A007318
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A013609 → basis A013610 → x ↦ x+1
• A135065 → x ↦ x+1 → x ↦ x/(x+1)
• A156992 → basis A007318 → basis A013609
• A134058 → basis A013610
• A178820 → basis A013610
• A085880 → basis A013609 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013609
• A356546 → basis A013611 → basis A007318
• A028326 → basis A007318 → basis A013609
• A178820 → basis A013610 → D
• basis-map coefficients of A113214, shifted → D → basis A013610
• A038243 → basis A013611
• A356546 → x ↦ x+1 → basis A013609
• A376827 → basis A013610
• basis-map coefficients of A112227 → x ↦ x+1 → basis A013610
• A132200 → basis A013610
• A178822 → basis A013610
• A178821 → basis A013611 → basis A007318
• A027466 → basis A013611 → x ↦ x/(x+1)
• A196347 → basis A013610 → D
• A028326 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A134346 → basis A013610
• A120910 → basis A013609
• A038208 → D → basis A013610
• A196347 → basis A013611 → x ↦ x+1
• A120910 → basis A013609 → D
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013609
• A135065 → x ↦ x+1 → basis A013609
• A156992 → basis A013609 → basis A013610
• A135089 → basis A013610
• A356546 → basis A013610
• A156992 → basis A013611 → x ↦ x+1
• A131431 → x ↦ x+1 → basis A013610
• A178821 → basis A013610 → reverse
• A085880 → basis A013611 → x ↦ x+1
• A085880 → basis A013610 → D
• A178822 → x ↦ x+1 → basis A013609
• A216973 → basis A013610
• A178821 → basis A013610 → D
• A120910 → basis A013609 → reverse
• A178822 → basis A013609 → basis A013610
• A028326 → basis A013611 → x ↦ x+1
• A136572 → basis A013609 → x ↦ x+1
• A178822 → basis A013611 → basis A007318
• A027465 → basis A013611 → basis A013610
• A085880 → basis A013610
• A156992 → basis A013610 → D
• A038330 → basis A013609 → x ↦ x/(x+1)
• A094305 → basis A013610
• A178820 → basis A007318 → basis A013609
• A038325 → basis A013610
• A094305 → x ↦ x+1 → basis A013609
• A038328 → basis A013609 → basis A013610
• A085881 → basis A013611 → basis A007318
• A085880 → D → basis A013610
• A038330 → basis A013609 → basis A013609
• A038338 → basis A013610 → D
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
• A120909 → basis A013610 → x ↦ x+1
• A003506 → x ↦ x+1 → basis A013609
• A038334 → D
• A178820 → basis A013609 → basis A013610
• basis-map coefficients of A127647 → basis A013610
• A038208 → basis A013610
• A013610 → basis A013611 → basis A013609
• A038221 → basis A013609 → 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.