A257609 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:          1;
  1:          4,          2;
  2:         24,         24,           4;
  3:        192,        288,         112,           8;
  4:       1920,       3840,        2400,         480,         16;
  5:      23040,      57600,       49920,       17280,       1984,         32;
  6:     322560,     967680,     1075200,      537600,     115584,       8064,        64;
  7:    5160960,   18063360,    24514560,    16128000,    5225472,     741888,     32512,      128;
  8:   92897280,  371589120,   596090880,   487710720,  213534720,   47738880,   4646400,   130560,    256;
  9: 1857945600, 8360755200, 15482880000, 15173222400, 8414945280, 2612736000, 419082240, 28661760, 523264, 512;

The row polynomials satisfy P(n,x) = (2*n+2+2*n*x)*P(n-1,x) - (2*x+2*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[(2*n + 2 + 2*n*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 2 + 2*n*x)*v[i - 1] - (2*x + 2*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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
    • 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.?
        • ↑ 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 ?

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