A028326 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:     2;
  1:    12,      8;
  2:    40,     96,      32;
  3:   112,    480,     576,     128;
  4:   288,   1792,    3840,    3072,      512;
  5:   704,   5760,   17920,   25600,    15360,     2048;
  6:  1664,  16896,   69120,  143360,   153600,    73728,     8192;
  7:  3840,  46592,  236544,  645120,  1003520,   860160,   344064,    32768;
  8:  8704, 122880,  745472, 2523136,  5160960,  6422528,  4587520,  1572864,  131072;
  9: 19456, 313344, 2211840, 8945664, 22708224, 37158912, 38535168, 23592960, 7077888, 524288;

The row polynomials satisfy (2*n-1+2*x)*P(n,x) = (4*n+2+(8*n+8)*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2.

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 2 + (8*n + 8)*x + 8*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 2 + (8*n + 8)*x + 8*x^2)*v[i - 1])/(2*n - 1 + 2*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 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_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • 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
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • Geometric basis map (elementary) polytool ? step ›

          • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A178820 → basis A013611 → D+1
• basis-map coefficients of A112227 → basis A013609 → D+1
• A130123 → basis A013609 → D+1
• basis-map coefficients of A026729 → basis A013609 → D+1
• A038208 → basis A013611 → D+1
• A156992 → basis A013611 → D+1
• A178822 → basis A013611 → D+1
• basis-map coefficients of A030528, shifted → basis A013609 → D+1
• A013609 → D → D+1
• A120909 → basis A013610 → D+1
• A136572 → basis A013609 → D+1
• A013609 → basis A013610 → D+1
• A085880 → basis A013611 → D+1
• A178821 → basis A013611 → D+1
• A196347 → basis A013611 → D+1
• A013609 → D+1
• A356546 → basis A013611 → D+1
• A085881 → basis A013611 → D+1
• A134082 → basis A013609
• A120909 → D+1
• A038221 → basis A013611 → D+1
• basis-map coefficients of A026729, shifted → basis A013609 → D+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.