A028338 → basis A013609 → basis A008277

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Submission text ?

Triangle begins:
  0:          1;
  1:          4,          2;
  2:         24,         24,           4;
  3:        192,        288,          96,          8;
  4:       1920,       3840,        1920,        320,         16;
  5:      23040,      57600,       38400,       9600,        960,        32;
  6:     322560,     967680,      806400,     268800,      40320,      2688,       64;
  7:    5160960,   18063360,    18063360,    7526400,    1505280,    150528,     7168,     128;
  8:   92897280,  371589120,   433520640,  216760320,   54190080,   7225344,   516096,   18432,   256;
  9: 1857945600, 8360755200, 11147673600, 6502809600, 1950842880, 325140480, 30965760, 1658880, 46080, 512;

The row polynomials satisfy P(n,x) = (2*n+2+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + 2 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = (2*n + 2 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
                • ↑ Liu–Wang theorem (above)
                • ↑ Iterated real-rootedness preservers (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 polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A271703: Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0)…
• A225470 → basis A013610 → basis A008277
• A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
• A130534 → basis A008277
• A089231 → D
• A089231: Triangular array A066667 or A008297 unsigned and transposed.
• A132393 → basis A008277 → D
• A271703 → reverse
• A225471 → basis A013611 → basis A008277
• A105278 → reverse

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.