A266213 → (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: 1;
  2: 1,  8;
  3: 1, 16,  18;
  4: 1, 24,  72,   32;
  5: 1, 32, 162,  192,    50;
  6: 1, 40, 288,  608,   400,    72;
  7: 1, 48, 450, 1408,  1650,   720,   98;
  8: 1, 56, 648, 2720,  4800,  3672, 1176,  128;
  9: 1, 64, 882, 4672, 11250, 12960, 7154, 1792, 162;

The row polynomials satisfy (n^2+(6*n^3-4*n^2+1)*x+(4*n^4+4*n^2+4*n-5)*x^2+(2*n^3+2*n^2-2*n-5)*x^3+(n^2-2*n+1)*x^4)*P(n,x) = ((n^2+2*n+1)*x+(6*n^3+14*n^2+10*n+3)*x^2+(4*n^4+16*n^3+28*n^2+28*n+7)*x^3+(2*n^3+8*n^2+8*n-3)*x^4+n^2*x^5)*P(n-2,x) + (n^2+(6*n^3+3*n^2-2*n)*x+(4*n^4+14*n^3+4*n^2+10*n+8)*x^2+(4*n^4+10*n^3+24*n^2+30*n+12)*x^3+(2*n^3+7*n^2+10*n-4)*x^4+n^2*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 + 2*n + 1)*x + (6*n^3 + 14*n^2 + 10*n + 3)*x^2 + (4*n^4 + 16*n^3 + 28*n^2 + 28*n + 7)*x^3 + (2*n^3 + 8*n^2 + 8*n - 3)*x^4 + n^2*x^5)*P[n - 2, x] + (n^2 + (6*n^3 + 3*n^2 - 2*n)*x + (4*n^4 + 14*n^3 + 4*n^2 + 10*n + 8)*x^2 + (4*n^4 + 10*n^3 + 24*n^2 + 30*n + 12)*x^3 + (2*n^3 + 7*n^2 + 10*n - 4)*x^4 + n^2*x^5)*P[n - 1, x])/(n^2 + (6*n^3 - 4*n^2 + 1)*x + (4*n^4 + 4*n^2 + 4*n - 5)*x^2 + (2*n^3 + 2*n^2 - 2*n - 5)*x^3 + (n^2 - 2*n + 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 + 2*n + 1)*x + (6*n^3 + 14*n^2 + 10*n + 3)*x^2 + (4*n^4 + 16*n^3 + 28*n^2 + 28*n + 7)*x^3 + (2*n^3 + 8*n^2 + 8*n - 3)*x^4 + n^2*x^5)*v[i - 2] + (n^2 + (6*n^3 + 3*n^2 - 2*n)*x + (4*n^4 + 14*n^3 + 4*n^2 + 10*n + 8)*x^2 + (4*n^4 + 10*n^3 + 24*n^2 + 30*n + 12)*x^3 + (2*n^3 + 7*n^2 + 10*n - 4)*x^4 + n^2*x^5)*v[i - 1])/(n^2 + (6*n^3 - 4*n^2 + 1)*x + (4*n^4 + 4*n^2 + 4*n - 5)*x^2 + (2*n^3 + 2*n^2 - 2*n - 5)*x^3 + (n^2 - 2*n + 1)*x^4)); 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-5c34cb2e6d8d80e7
• 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-5c34cb2e6d8d80e7
• 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-5c34cb2e6d8d80e7
• 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-5c34cb2e6d8d80e7, e-dcf5be93843f2cf7
• 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-dcf5be93843f2cf7
• 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-dcf5be93843f2cf7
• 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-dcf5be93843f2cf7

Proof chains

2 incoming proof steps. 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-5c34cb2e6d8d80e7

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

  • premise (sequence): Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by u…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-1e993b9b3c7faf34 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

• preserver_image euler_polynomial, shift 0 e-dcf5be93843f2cf7

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

  • premise (sequence): Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-f5199491b1877911 root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

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.