A356546 → x ↦ x+1 → θ+1

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Submission text ?

Triangle begins:
  0:        1;
  1:        4,         4;
  2:       24,        48,        18;
  3:      160,       480,       360,         80;
  4:     1120,      4480,      5040,       2240,       350;
  5:     8064,     40320,     60480,      40320,     12600,      1512;
  6:    59136,    354816,    665280,     591360,    277200,     66528,      6468;
  7:   439296,   3075072,   6918912,    7687680,   4804800,   1729728,    336336,    27456;
  8:  3294720,  26357760,  69189120,   92252160,  72072000,  34594560,  10090080,  1647360,  115830;
  9: 24893440, 224040960, 672122880, 1045524480, 980179200, 588107520, 228708480, 56010240, 7876440, 486200;

The row polynomials satisfy (2*n+n^2*x)*P(n,x) = (16*n-8+(8*n^2+12*n-8)*x+(4*n^2+2*n-2)*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[((16*n - 8 + (8*n^2 + 12*n - 8)*x + (4*n^2 + 2*n - 2)*x^2)*P[n - 1, x])/(2*n + n^2*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] = ((16*n - 8 + (8*n^2 + 12*n - 8)*x + (4*n^2 + 2*n - 2)*x^2)*v[i - 1])/(2*n + n^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 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_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (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 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
• A028326 → basis A007318 → θ+1
• A135065 → x ↦ x+1 → θ+1
• image under θ+1 → θ+1 → D+1 → basis A013609
• A094305 → x ↦ x+1 → θ+1
• A038338 → x ↦ x+1 → θ+1
• A120910 → D → θ+1
• A027466 → basis A013611 → θ+1
• A178822 → x ↦ x+1 → θ+1
• A156992 → basis A007318 → θ+1
• A196347 → x ↦ x+1 → θ+1
• A085880 → x ↦ x+1 → θ+1
• A131431 → D+1 → basis A013609
• A178820 → basis A007318 → θ+1
• A038208 → basis A007318 → θ+1
• A038207 → θ+1
• A132440 → D+1 → basis A013609
• A038330 → basis A013609 → θ+1
• A085881 → basis A007318 → θ+1
• A120910 → θ+1
• A038221 → x ↦ x+1 → θ+1
• A003506 → x ↦ x+1 → θ+1
• A038325 → basis A007318 → θ+1
• A178821 → basis A007318 → θ+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.