A085880 → basis A013610 → D

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Triangle begins:
   1:        0;
   2:        3;
   3:       24,        36;
   4:      180,       540,        405;
   5:     1344,      6048,       9072,       4536;
   6:    10080,     60480,     136080,     136080,       51030;
   7:    76032,    570240,    1710720,    2566080,     1924560,      577368;
   8:   576576,   5189184,   19459440,   38918880,    43783740,    26270244,     6567561;
   9:  4392960,  46126080,  207567360,  518918400,   778377600,   700539840,   350269920,   75057840;
  10: 33606144, 403273728, 2117187072, 6351561216, 11909177280, 14291012736, 10718259552, 4593539808, 861288714;

The row polynomials satisfy (n^2-2*n)*P(n,x) = (8*n^2-20*n+12+(12*n^2-30*n+18)*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 20*n + 12 + (12*n^2 - 30*n + 18)*x)*P[n - 1, x])/(n^2 - 2*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = ((8*n^2 - 20*n + 12 + (12*n^2 - 30*n + 18)*x)*v[i - 1])/(n^2 - 2*n)); 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 (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
  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) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan nu…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
        • 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) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • 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 T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (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) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan nu…?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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