A156992 → basis A013610 → D

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Triangle begins:
  0:           0;
  1:           6;
  2:          72,          108;
  3:         864,         2592,          1944;
  4:       11520,        51840,         77760,         38880;
  5:      172800,      1036800,       2332800,       2332800,        874800;
  6:     2903040,     21772800,      65318400,      97977600,      73483200,       22044960;
  7:    54190080,    487710720,    1828915200,    3657830400,    4115059200,     2469035520,     617258880;
  8:  1114767360,  11705057280,   52672757760,  131681894400,  197522841600,   177770557440,   88885278720,   19046845440;
  9: 25082265600, 300987187200, 1580182732800, 4740548198400, 8888527872000, 10666233446400, 7999675084800, 3428432179200, 642831033600;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 2*n + (3*n^2 + 3*n)*x)*v[i - 1])/(n - 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

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

  • from basis_map_A013610 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_A007318 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)
        • 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)
            • ↑ 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)

Related ?

Same fingerprint ?

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