A156992 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
  0:              1;
  1:             12,              16;
  2:            216,             576,              384;
  3:           5184,           20736,            27648,            12288;
  4:         155520,          829440,          1658880,          1474560,            491520;
  5:        5598720,        37324800,         99532800,        132710400,          88473600,          23592960;
  6:      235146240,      1881169920,       6270566400,      11147673600,       11147673600,        5945425920,        1321205760;
  7:    11287019520,    105345515520,     421382062080,     936404582400,     1248539443200,      998831554560,      443925135360,      84557168640;
  8:   609499054080,   6501323243520,   30339508469760,   80905355919360,   134842259865600,   143831743856640,    95887829237760,   36528696852480,    6088116142080;
  9: 36569943244800, 438839318937600, 2340476367667200, 7281482032742400, 14562964065484800, 19417285420646400, 17259809262796800, 9862748150169600, 3287582716723200, 487049291366400;

The row polynomials satisfy P(n,x) = (6*n+6+(8*n+8)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6*n + 6 + (8*n + 8)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6*n + 6 + (8*n + 8)*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 - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
        • 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

                  • from Triangle read by rows: row n consists of n zeros followed by n!.?
                    • 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
                    • Parent-row provers of polytool polytool ? step › direct check
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from Triangle read by rows: row n consists of n zeros followed by n!.?
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Parent-row provers of polytool (above)
                        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                          • from Triangle read by rows: row n consists of n zeros followed by n!.?
                            • ↑ Iterated real-rootedness preservers (above)
                            • ↑ Parent-row provers of polytool (above)
                            • ↑ Parent-row provers of polytool (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

                  • from Triangle read by rows: row n consists of n zeros followed by n!.?
                    • ↑ Iterated real-rootedness preservers (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                    • ↑ Laguerre’s theorem on multiplier sequences (above)
                    • ↑ Laguerre’s theorem on multiplier sequences (above)
            • 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 theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                • ↑ Laguerre’s theorem on multiplier sequences (above)
                • ↑ Laguerre’s theorem on multiplier sequences (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (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_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Geometric basis map (elementary) (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ Geometric basis map (elementary) (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A356546 → x ↦ x+1 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A038335 → reverse
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A085881 → basis A013611 → basis A013609
• A013610 → basis A007318
• A038334 → basis A013609 → basis A013610
• A120910 → D → basis A013611
• A178822 → basis A013611 → basis A013609
• A038329 → basis A013610 → basis A013609
• A003506 → x ↦ x+1 → basis A013611
• A178822 → x ↦ x+1 → basis A013611
• A120909 → basis A013610 → basis A013609
• A028326 → basis A007318 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A356546 → basis A013611 → basis A013609
• A120910 → basis A013611 → reverse
• A196347 → basis A013611 → basis A013609
• A027465 → D → basis A013610
• A038335 → D
• A013609 → D → basis A013609
• A085880 → x ↦ x+1 → basis A013611
• A094305 → x ↦ x+1 → basis A013611
• A013610 → basis A013611 → x ↦ x+1
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• A196347 → x ↦ x+1 → basis A013611
• A038330 → basis A013610 → reverse
• A038329 → basis A013611 → basis A013610
• A028326 → basis A013611 → basis A013609
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A038243 → basis A013609 → x ↦ x/(x+1)
• A120910 → basis A013611 → D
• A038284 → reverse
• A038284 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A038330 → basis A013610
• A013610 → basis A007318 → D
• A178820 → basis A013611 → basis A013609
• A130123 → basis A013609 → basis A013609
• A085880 → basis A013611 → basis A013609
• A120910 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• A178821 → basis A013611 → basis A013609
• A038279 → basis A013610 → x ↦ x/(x+1)
• A038221 → basis A013611 → basis A013609
• A130123 → basis A013610 → x ↦ x+1
• A135065 → x ↦ x+1 → basis A013611
• A038208 → basis A013611 → basis A013609
• A136572 → basis A013610 → basis A007318
• A178821 → basis A007318 → basis A013611
• A038330 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A178820 → basis A007318 → basis A013611
• A038330 → basis A013610 → D
• A013609 → basis A013609
• A027465 → x ↦ x/(x+1)
• A013609 → basis A013609 → 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. [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.