A173018 → x ↦ x+1 → (θ+1)²

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Submission text ?

Triangle begins:
   1:      1;
   2:      1;
   3:      2,       4;
   4:      6,      24,        9;
   5:     24,     144,      126,       16;
   6:    120,     960,     1350,      480,       25;
   7:    720,    7200,    14040,     8640,     1550,      36;
   8:   5040,   60480,   151200,   134400,    45150,    4536,     49;
   9:  40320,  564480,  1723680,  2016000,  1020600,  208656,  12446,    64;
  10: 362880, 5806080, 20956320, 30481920, 20853000, 6713280, 889350, 32640, 81;

The row polynomials satisfy (n^2-4*n+4)*P(n,x) = -(n^4-6*n^3+13*n^2-12*n+4+(4*n^4-32*n^3+96*n^2-128*n+60)*x-(9*n^4-74*n^3+221*n^2-284*n+132)*x^2)*P(n-2,x) + ((2*n^3-11*n^2+20*n-11)*x+(4*n^4-28*n^3+68*n^2-64*n+21)*x^2+(4*n^4-46*n^3+167*n^2-244*n+124)*x^3)*d/dx P(n-2,x) - ((n^2-4*n+3)*x^2+(4*n^3-20*n^2+32*n-17)*x^3+(4*n^3-21*n^2+36*n-20)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^3-9*n^2+13*n-6+(4*n^3-24*n^2+48*n-32)*x)*P(n-1,x) - ((2*n^2-8*n+7)*x+(4*n^3-20*n^2+32*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 - 6*n^3 + 13*n^2 - 12*n + 4 + (4*n^4 - 32*n^3 + 96*n^2 - 128*n + 60)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*P[n - 2, x] + ((2*n^3 - 11*n^2 + 20*n - 11)*x + (4*n^4 - 28*n^3 + 68*n^2 - 64*n + 21)*x^2 + (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*D[P[n - 2, x], x] - ((n^2 - 4*n + 3)*x^2 + (4*n^3 - 20*n^2 + 32*n - 17)*x^3 + (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^3 - 9*n^2 + 13*n - 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*P[n - 1, x] - ((2*n^2 - 8*n + 7)*x + (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*D[P[n - 1, x], x])/(n^2 - 4*n + 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n^4 - 6*n^3 + 13*n^2 - 12*n + 4 + (4*n^4 - 32*n^3 + 96*n^2 - 128*n + 60)*x - (9*n^4 - 74*n^3 + 221*n^2 - 284*n + 132)*x^2)*v[i - 2] + ((2*n^3 - 11*n^2 + 20*n - 11)*x + (4*n^4 - 28*n^3 + 68*n^2 - 64*n + 21)*x^2 + (4*n^4 - 46*n^3 + 167*n^2 - 244*n + 124)*x^3)*deriv(v[i - 2], x) - ((n^2 - 4*n + 3)*x^2 + (4*n^3 - 20*n^2 + 32*n - 17)*x^3 + (4*n^3 - 21*n^2 + 36*n - 20)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^3 - 9*n^2 + 13*n - 6 + (4*n^3 - 24*n^2 + 48*n - 32)*x)*v[i - 1] - ((2*n^2 - 8*n + 7)*x + (4*n^3 - 20*n^2 + 32*n - 16)*x^2)*deriv(v[i - 1], x))/(n^2 - 4*n + 4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from 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 shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
        • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
        • Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? 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 numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.?
        • ↑ Ma–Wang interval theorem (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A090582 → (θ+1)²
• A008292 → x ↦ x+1 → (θ+1)²

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.