A249250 → (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,  12;
  2: 1,  24,   27;
  3: 1,  36,  135,    48;
  4: 1,  48,  324,   384,    75;
  5: 1,  60,  594,  1440,   825,    108;
  6: 1,  72,  945,  3648,  4275,   1512,    147;
  7: 1,  84, 1377,  7440, 14475,  10044,   2499,   192;
  8: 1,  96, 1890, 13248, 37500,  43200,  20286,  3840,  243;
  9: 1, 108, 2484, 21504, 81450, 138888, 106428, 36864, 5589, 300;

The row polynomials satisfy (n^2+2*n+1+(9*n^3+23*n^2+18*n+5)*x+(9*n^4+42*n^3+78*n^2+75*n+23)*x^2+(3*n^3+14*n^2+13*n-5)*x^3+n^2*x^4)*P(n,x) = ((2*n^2+8*n+8)*x+(18*n^3+100*n^2+182*n+110)*x^2+(18*n^4+156*n^3+516*n^2+786*n+454)*x^3+(6*n^3+46*n^2+100*n+50)*x^4+(2*n^2+4*n+2)*x^5)*P(n-2,x) + (n^2+2*n+1+(9*n^3+33*n^2+34*n+9)*x+(9*n^4+69*n^3+161*n^2+160*n+78)*x^2+(9*n^4+63*n^3+197*n^2+328*n+222)*x^3+(3*n^3+21*n^2+50*n+25)*x^4+(n^2+2*n+1)*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.

P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n^2 + 8*n + 8)*x + (18*n^3 + 100*n^2 + 182*n + 110)*x^2 + (18*n^4 + 156*n^3 + 516*n^2 + 786*n + 454)*x^3 + (6*n^3 + 46*n^2 + 100*n + 50)*x^4 + (2*n^2 + 4*n + 2)*x^5)*P[n - 2, x] + (n^2 + 2*n + 1 + (9*n^3 + 33*n^2 + 34*n + 9)*x + (9*n^4 + 69*n^3 + 161*n^2 + 160*n + 78)*x^2 + (9*n^4 + 63*n^3 + 197*n^2 + 328*n + 222)*x^3 + (3*n^3 + 21*n^2 + 50*n + 25)*x^4 + (n^2 + 2*n + 1)*x^5)*P[n - 1, x])/(n^2 + 2*n + 1 + (9*n^3 + 23*n^2 + 18*n + 5)*x + (9*n^4 + 42*n^3 + 78*n^2 + 75*n + 23)*x^2 + (3*n^3 + 14*n^2 + 13*n - 5)*x^3 + n^2*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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (((2*n^2 + 8*n + 8)*x + (18*n^3 + 100*n^2 + 182*n + 110)*x^2 + (18*n^4 + 156*n^3 + 516*n^2 + 786*n + 454)*x^3 + (6*n^3 + 46*n^2 + 100*n + 50)*x^4 + (2*n^2 + 4*n + 2)*x^5)*v[i - 2] + (n^2 + 2*n + 1 + (9*n^3 + 33*n^2 + 34*n + 9)*x + (9*n^4 + 69*n^3 + 161*n^2 + 160*n + 78)*x^2 + (9*n^4 + 63*n^3 + 197*n^2 + 328*n + 222)*x^3 + (3*n^3 + 21*n^2 + 50*n + 25)*x^4 + (n^2 + 2*n + 1)*x^5)*v[i - 1])/(n^2 + 2*n + 1 + (9*n^3 + 23*n^2 + 18*n + 5)*x + (9*n^4 + 42*n^3 + 78*n^2 + 75*n + 23)*x^2 + (3*n^3 + 14*n^2 + 13*n - 5)*x^3 + n^2*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-5c8788bb6f0033f7
• 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-5c8788bb6f0033f7
• 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-5c8788bb6f0033f7
• 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-5c8788bb6f0033f7

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-5c8788bb6f0033f7

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

  • premise (sequence): Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
    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-86238c6a50aeb955 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.