A126351 → (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,   8;
  2: 1,  20,   36;
  3: 1,  36,  171,    128;
  4: 1,  56,  495,   1040,     400;
  5: 1,  80, 1125,   4560,    5275,    1152;
  6: 1, 108, 2205,  14560,   33775,   23940,     3136;
  7: 1, 140, 3906,  38080,  147525,  218484,   100891,    8192;
  8: 1, 176, 6426,  86688,  504525, 1280664,  1290415,  403520,   20736;
  9: 1, 216, 9990, 178080, 1452675, 5639760, 10006045, 7145280, 1552851, 51200;

The row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(n^2+3*n+2+(4*n^3+12*n^2+12*n)*x-(9*n^4+34*n^3+41*n^2+16*n)*x^2)*P(n-2,x) - (x-(4*n^3+16*n^2+20*n+13)*x^2-(4*n^4+2*n^3-31*n^2-52*n-23)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3+15*n^2+18*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+5*n+3+(4*n^3+12*n^2+12*n+4)*x)*P(n-1,x) + (x-(4*n^3+16*n^2+20*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*P[n - 2, x] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*P[n - 1, x] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*v[i - 2] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*v[i - 1] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); 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-e293f6e7721f27fd
• 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-e293f6e7721f27fd
• 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-e293f6e7721f27fd
• 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-e293f6e7721f27fd

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-e293f6e7721f27fd

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

  • premise (sequence): Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-0c08584d383e1215 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.