A109692 → basis A013610 → (θ+1)²

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:         1;
  1:         2,          12;
  2:         8,         120,          243;
  3:        48,        1200,         5508,          6480;
  4:       384,       13632,       100764,        257472,         212625;
  5:      3840,      177792,      1835784,       7411392,       12988350,        8266860;
  6:     46080,     2640384,     35230464,     196636032,      538010100,      716408712,       371319795;
  7:    645120,    44153856,    724920192,    5195549952,    19514649600,    40244369184,     43227839592,     18914575680;
  8:  10321920,   822583296,  16069300992,  141122414592,   677546499600,  1908459201024,   3156613045992,   2843371537920,   1077244193025;
  9: 185794560, 16912171008, 383638851072, 3997153419264, 23441529405600, 84111280549056, 189297911032272, 261449768950272, 202921142711130, 67826486227500;

The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(96*n^3-288*n^2+264*n-72)*x-(648*n^3-2268*n^2+2430*n-729)*x^2)*P(n-2,x) + ((96*n^3-288*n^2+264*n-72)*x^2+(288*n^3-1008*n^2+1080*n-324)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+2+(48*n^2-96*n+36)*x)*P(n-1,x) - (48*n^2-96*n+36)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.

P[0, x_] := 1;
P[1, x_] := 2 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (96*n^3 - 288*n^2 + 264*n - 72)*x - (648*n^3 - 2268*n^2 + 2430*n - 729)*x^2)*P[n - 2, x] + ((96*n^3 - 288*n^2 + 264*n - 72)*x^2 + (288*n^3 - 1008*n^2 + 1080*n - 324)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 2 + (48*n^2 - 96*n + 36)*x)*P[n - 1, x] - (48*n^2 - 96*n + 36)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (96*n^3 - 288*n^2 + 264*n - 72)*x - (648*n^3 - 2268*n^2 + 2430*n - 729)*x^2)*v[i - 2] + ((96*n^3 - 288*n^2 + 264*n - 72)*x^2 + (288*n^3 - 1008*n^2 + 1080*n - 324)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 2 + (48*n^2 - 96*n + 36)*x)*v[i - 1] - (48*n^2 - 96*n + 36)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).?
    • Geometric basis map (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 [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 (above)
                • 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 polytool ? step › direct check
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)
    • Geometric basis map (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 (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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.