A085881 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
  0:           1;
  1:           2,            8;
  2:          12,           96,           144;
  3:         120,         1440,          4320,           3840;
  4:        1680,        26880,        120960,         215040,          134400;
  5:       30240,       604800,       3628800,        9676800,        12096000,         5806080;
  6:      665280,     15966720,     119750400,      425779200,       798336000,       766402560,       298045440;
  7:    17297280,    484323840,    4358914560,    19372953600,     48432384000,     69742632960,     54244270080,     17712414720;
  8:   518918400,  16605388800,  174356582400,   929901772800,   2905943040000,   5579410636800,   6509312409600,   4250979532800,   1195587993600;
  9: 17643225600, 635156121600, 7621873459200, 47424990412800, 177843714048000, 426824913715200, 663949865779200, 650399868518400, 365849926041600, 90333315072000;

The row polynomials satisfy (1+2*n*x)*P(n,x) = (4*n-2+(8*n^2+12*n-8)*x+(16*n^2+8*n-8)*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[((4*n - 2 + (8*n^2 + 12*n - 8)*x + (16*n^2 + 8*n - 8)*x^2)*P[n - 1, x])/(1 + 2*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] = ((4*n - 2 + (8*n^2 + 12*n - 8)*x + (16*n^2 + 8*n - 8)*x^2)*v[i - 1])/(1 + 2*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_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • 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
        • 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
        • 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 A001147(n).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (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) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
            • ↑ Geometric basis map (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 A001147(n).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A196347 → basis A013611 → θ+1
• basis-map coefficients of A112227 → basis A013609 → θ+1
• A116666 → x ↦ x/(x+1)
• A013609 → θ+1
• A130123 → basis A013609 → θ+1
• A028326 → basis A013611 → θ+1
• A178822 → basis A013611 → θ+1
• A178821 → basis A013611 → θ+1
• A085880 → basis A013611 → θ+1
• A013609 → basis A013610 → θ+1
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1
• A120909 → basis A013610 → θ+1
• A038208 → basis A013611 → θ+1
• A038221 → basis A013611 → θ+1
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
• A013609 → D → θ+1
• A178820 → basis A013611 → θ+1
• A156992 → basis A013611 → θ+1
• A136572 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A356546 → basis A013611 → θ+1
• A120909 → θ+1
• A134082 → basis A007318 → x ↦ x+1
• A116666 → reverse → x ↦ x+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.