A356546 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
  0:        1;
  1:        4,        16;
  2:       24,       192,         288;
  3:      160,      1920,        5760,        5120;
  4:     1120,     17920,       80640,      143360,        89600;
  5:     8064,    161280,      967680,     2580480,      3225600,      1548288;
  6:    59136,   1419264,    10644480,    37847040,     70963200,     68124672,     26492928;
  7:   439296,  12300288,   110702592,   492011520,   1230028800,   1771241472,   1377632256,    449839104;
  8:  3294720, 105431040,  1107025920,  5904138240,  18450432000,  35424829440,  41328967680,  26990346240,   7591034880;
  9: 24893440, 896163840, 10753966080, 66913566720, 250925875200, 602222100480, 936789934080, 917671772160, 516190371840, 127454412800;

The row polynomials satisfy (n+2*n^2*x)*P(n,x) = (8*n-4+(16*n^2+24*n-16)*x+(32*n^2+16*n-16)*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[((8*n - 4 + (16*n^2 + 24*n - 16)*x + (32*n^2 + 16*n - 16)*x^2)*P[n - 1, x])/(n + 2*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] = ((8*n - 4 + (16*n^2 + 24*n - 16)*x + (32*n^2 + 16*n - 16)*x^2)*v[i - 1])/(n + 2*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_A013611 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 basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 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 (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ 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)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038221 → basis A013611 → θ+1
• A178820 → basis A013611 → θ+1
• A196347 → basis A013611 → θ+1
• A028326 → basis A013611 → θ+1
• A013609 → basis A013610 → θ+1
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
• A134082 → basis A007318 → x ↦ x+1
• A116666 → reverse → x ↦ x+1
• A038208 → basis A013611 → θ+1
• A130123 → basis A013609 → θ+1
• A120909 → basis A013610 → θ+1
• A013609 → D → θ+1
• basis-map coefficients of A030528, shifted → basis A013609 → θ+1
• A136572 → basis A013609 → θ+1
• A178821 → basis A013611 → θ+1
• A178822 → basis A013611 → θ+1
• A013609 → θ+1
• A156992 → basis A013611 → θ+1
• A120909 → θ+1
• basis-map coefficients of A026729 → basis A013609 → θ+1
• A085880 → basis A013611 → θ+1
• A085881 → basis A013611 → θ+1
• A116666 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609 → 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.