A039755 → (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:
   1: 1;
   2: 1,     4;
   3: 1,    16,       9;
   4: 1,    52,      81,       16;
   5: 1,   160,     522,      256,       25;
   6: 1,   484,    2970,     2720,      625,      36;
   7: 1,  1456,   15939,    24320,     9875,    1296,      49;
   8: 1,  4372,   82971,   198576,   126875,   28476,    2401,     64;
   9: 1, 13120,  424692,  1537536,  1452150,  495936,   69972,   4096,   81;
  10: 1, 39364, 2152980, 11517760, 15471750, 7546392, 1584660, 152832, 6561, 100;

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

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[(1 - 4*x + 9*x^2)*P[n - 2, x] - (4*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (4*x^2 - 8*x^3)*D[P[n - 2, x], {x, 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, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (1 - 4*x + 9*x^2)*v[i - 2] - (4*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (4*x^2 - 8*x^3)*deriv(deriv(v[i - 2], x), x) + 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-c92588563eed0abe
• Coefficients of one sign. Every nonzero row, n >= 1: 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-c92588563eed0abe
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-c92588563eed0abe
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-c92588563eed0abe

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-c92588563eed0abe

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

  • premise (sequence): Triangle of B-analogs of Stirling numbers of the second kind.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c873e8cf4f19d199 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.