A257620 → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           1;
  1:           6,            3;
  2:          54,           54,            9;
  3:         648,          972,          378,           27;
  4:        9720,        19440,        12150,         2430,           81;
  5:      174960,       437400,       379080,       131220,        15066,          243;
  6:     3674160,     11022480,     12247200,      6123600,      1316574,        91854,         729;
  7:    88179840,    308629440,    418854240,    275562000,     89282088,     12675852,      555498,       2187;
  8:  2380855680,   9523422720,  15277157280,  12499492320,   5472661320,   1223495280,   119082150,    3346110,     6561;
  9: 71425670400, 321415516800, 595213920000, 583309641600, 323498765520, 100442349000, 16110929160, 1101854340, 20116026, 19683;

The row polynomials satisfy P(n,x) = (3*n+3+3*n*x)*P(n-1,x) - (3*x+3*x^2)*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[(3*n + 3 + 3*n*x)*P[n - 1, x] - (3*x + 3*x^2)*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] = (3*n + 3 + 3*n*x)*v[i - 1] - (3*x + 3*x^2)*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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A257609 → x ↦ x+1
• A173018 → x ↦ x+1
• row-factor cofactor of A101842 → x ↦ x/(x+1)
• A131689: 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.
• A008292 → x ↦ x+1
• A163626 → x ↦ x+1
• A028246 → θ+1
• A257609 → x ↦ x/(x+1)
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
• A173018 → reverse (shift +1) → x ↦ x/(x+1)
• A173018 → reverse (shift +1) → reverse → x ↦ x+1
• A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1…
• A257620 → x ↦ x/(x+1)

References

1. [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