A156992 → basis A013609 → θ+1

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Triangle begins:
  0:          1;
  1:          4,           8;
  2:         24,          96,           72;
  3:        192,        1152,         1728,          768;
  4:       1920,       15360,        34560,        30720,          9600;
  5:      23040,      230400,       691200,       921600,        576000,        138240;
  6:     322560,     3870720,     14515200,     25804800,      24192000,      11612160,       2257920;
  7:    5160960,    72253440,    325140480,    722534400,     903168000,     650280960,     252887040,     41287680;
  8:   92897280,  1486356480,   7803371520,  20808990720,   32514048000,   31213486080,   18207866880,   5945425920,    836075520;
  9: 1857945600, 33443020800, 200658124800, 624269721600, 1170505728000, 1404606873600, 1092472012800, 535088332800, 150493593600, 18579456000;

The row polynomials satisfy (1+n*x)*P(n,x) = (2*n+2+(2*n^2+6*n+4)*x+(2*n^2+4*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (2*n^2 + 6*n + 4)*x + (2*n^2 + 4*n + 2)*x^2)*P[n - 1, x])/(1 + n*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] = ((2*n + 2 + (2*n^2 + 6*n + 4)*x + (2*n^2 + 4*n + 2)*x^2)*v[i - 1])/(1 + 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_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) [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)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A196347 → basis A013609 → θ+1
• A131431 → x ↦ x+1 → θ+1
• A356546 → θ+1
• A156992 → θ+1
• A344565 → D+1
• A178820 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A085880 → basis A013609 → D+1
• A376827 → D+1
• basis-map coefficients of A127647 → θ+1
• A376827 → θ+1
• A135838 → θ+1
• A178821 → basis A013609 → D+1
• A156992 → basis A013609 → D+1
• A085880 → θ+1
• A134346 → θ+1
• A038325 → basis A013609 → θ+1
• A356546 → basis A013609 → θ+1
• A178822 → basis A013609 → D+1
• A085881 → basis A013609 → D+1
• A093375 → D → D
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A094305 → D+1
• A038325 → θ+1
• A178822 → basis A013609 → θ+1
• A027465 → basis A013611 → D+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A085881 → θ+1
• A178821 → basis A013609 → θ+1
• A196347 → D+1
• A085881 → basis A013609 → θ+1
• A038208 → D → D+1
• A038208 → D+1
• A178822 → θ+1
• A038221 → θ+1
• A038208 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A356546 → basis A013609 → D+1
• A028326 → basis A013609 → D+1
• A038330 → basis A013611 → θ+1
• A038338 → D+1
• A156992 → D+1
• A038338 → θ+1
• A178820 → θ+1
• A028326 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A038221 → basis A013609 → D+1
• A085880 → D → θ+1
• A038208 → D → θ+1
• A094305 → θ+1
• A028326 → D → D+1
• A027465 → basis A013611 → θ+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A093375 → D
• A178821 → θ+1
• A178820 → basis A013609 → D+1
• A131431 → x ↦ x+1 → D+1
• A135065 → θ+1
• A085880 → D → D+1
• A028326 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A085880 → D+1
• A178822 → D+1
• A120910 → basis A013610 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D → D+1
• A131431 → D+1 → basis A007318
• A038325 → D+1
• A120910 → basis A013610 → θ+1
• A038221 → D+1
• basis-map coefficients of A127647 → D+1
• A344565 → θ+1
• A028326 → θ+1
• A028326 → D → θ+1
• A216973 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A038338 → basis A013609 → θ+1
• A085881 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A178820 → D+1
• A038325 → basis A013609 → D+1
• A003506 → D → θ+1
• A134346 → D+1
• A038338 → basis A013609 → D+1
• A196347 → basis A013609 → D+1
• A178821 → D+1
• A038221 → basis A013609 → θ+1
• A038330 → basis A013611 → D+1
• A085880 → basis A013609 → θ+1
• A124860 → D+1
• A135838 → D+1
• A356546 → D+1
• A196347 → θ+1
• A135065 → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A124860 → 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.