A075497 → (θ+1)²

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:   1;
  1:   2,      4;
  2:   4,     24,        9;
  3:   8,    112,      108,       16;
  4:  16,    480,      900,      320,       25;
  5:  32,   1984,     6480,     4160,      750,       36;
  6:  64,   8064,    43344,    44800,    14000,     1512,      49;
  7: 128,  32512,   278208,   435456,   210000,    38304,    2744,     64;
  8: 256, 130560,  1742400,  3978240,  2780400,   762048,   90552,   4608,   81;
  9: 512, 523264, 10748160, 34923520, 34020000, 13148352, 2304960, 192000, 7290, 100;

The row polynomials satisfy P(n,x) = -(8*x-9*x^2)*P(n-2,x) - (8*x-8*x^2-4*x^3)*d/dx P(n-2,x) - (4*x^2-8*x^3)*d^2/dx^2 P(n-2,x) + (2+4*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(8*x - 9*x^2)*P[n - 2, x] - (8*x - 8*x^2 - 4*x^3)*D[P[n - 2, x], x] - (4*x^2 - 8*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 4*x)*P[n - 1, x] + (4*x - 4*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; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(8*x - 9*x^2)*v[i - 2] - (8*x - 8*x^2 - 4*x^3)*deriv(v[i - 2], x) - (4*x^2 - 8*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 4*x)*v[i - 1] + (4*x - 4*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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-d445ae4d7e7f6688
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-d445ae4d7e7f6688
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-d445ae4d7e7f6688
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-d445ae4d7e7f6688

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image euler_polynomial, shift 0 e-d445ae4d7e7f6688

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Stirling2 triangle with scaled diagonals (powers of 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca5f07049cdde3bf root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [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
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.