A085881 → basis A007318 → D+1

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Submission text ?

Triangle begins:
  0:           1;
  1:           3,            1;
  2:          24,           18,            3;
  3:         300,          360,          135,           15;
  4:        5040,         8400,         5040,         1260,          105;
  5:      105840,       226800,       189000,        75600,        14175,          945;
  6:     2661120,      6985440,      7484400,      4158000,      1247400,       187110,       10395;
  7:    77837760,    242161920,    317837520,    227026800,     94594500,     22702680,     2837835,     135135;
  8:  2594592000,   9340531200,  14529715200,  12713500800,   6810804000,   2270268000,   454053600,   48648600,   2027025;
  9: 97037740800, 396972576000, 714550636800, 741015475200, 486291405600, 208410602400, 57891834000, 9924314400, 930404475, 34459425;

The row polynomials satisfy (n+1+x)*P(n,x) = (4*n^2+6*n-4+(2*n^2+7*n-4)*x+(2*n-1)*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 + 6*n - 4 + (2*n^2 + 7*n - 4)*x + (2*n - 1)*x^2)*P[n - 1, x])/(n + 1 + 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 + 6*n - 4 + (2*n^2 + 7*n - 4)*x + (2*n - 1)*x^2)*v[i - 1])/(n + 1 + 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_A007318 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 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)

Related ?

Same fingerprint ?

• basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1
• A038338 → x ↦ x+1 → D+1
• A093375 → x ↦ x+1 → reverse
• A135065 → x ↦ x+1 → D+1
• A196347 → x ↦ x+1 → D+1
• A093375 → D → basis A007318
• A120910 → D+1
• A356546 → x ↦ x+1 → D+1
• A103406 → x ↦ x/(x+1)
• A038207 → D+1
• A027466 → basis A013611 → D+1
• A003506 → x ↦ x+1 → D+1
• A156992 → basis A007318 → D+1
• A038330 → basis A013609 → D+1
• A178820 → basis A007318 → D+1
• A178821 → basis A007318 → D+1
• A028326 → basis A007318 → D+1
• A094305 → x ↦ x+1 → D+1
• A085880 → x ↦ x+1 → D+1
• A038221 → x ↦ x+1 → D+1
• A178822 → x ↦ x+1 → D+1
• A038208 → basis A007318 → D+1
• A120910 → D → D+1
• A093375 → x ↦ x+1
• A038325 → basis A007318 → 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.