A154986 → θ+1

OEIS matches ?

• A001710: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A357711: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A188588: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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;
  2: 1,   8,     3;
  3: 1,  22,    33,       4;
  4: 1,  48,   210,      96,       5;
  5: 1,  90,   942,    1256,     225,       6;
  6: 1, 152,  3237,   10912,    5395,     456,       7;
  7: 1, 238,  9135,   67980,   84975,   18270,     833,      8;
  8: 1, 352, 22260,  321856,  933790,  482784,   51940,   1408,    9;
  9: 1, 498, 48492, 1228496, 7454310, 8945172, 2149868, 129312, 2241, 10;

The row polynomials satisfy (n^2-3*n+3)*P(n,x) = -(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2*P(n-4,x) - ((2*n^4-8*n^3+12*n^2-10*n+4)*x+(2*n^4-10*n^3+20*n^2-20*n+8)*x^2)*P(n-3,x) - (n^2-n+1-(2*n^4-8*n^3+10*n^2-4*n-2)*x+(n^2-3*n+3)*x^2)*P(n-2,x) + (2*n^2-4*n+4+(2*n^2-6*n+6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 8*x + 3*x^2, P(3,x) = 1 + 22*x + 33*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 8*x + 3*x^2;
P[3, x_] := 1 + 22*x + 33*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*P[n - 4, x] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*P[n - 3, x] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - 3*n + 3)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + 4 + (2*n^2 - 6*n + 6)*x)*P[n - 1, x])/(n^2 - 3*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 8*x + 3*x^2; v[4] = 1 + 22*x + 33*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2*v[i - 4] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (2*n^4 - 10*n^3 + 20*n^2 - 20*n + 8)*x^2)*v[i - 3] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 10*n^2 - 4*n - 2)*x + (n^2 - 3*n + 3)*x^2)*v[i - 2] + (2*n^2 - 4*n + 4 + (2*n^2 - 6*n + 6)*x)*v[i - 1])/(n^2 - 3*n + 3)); 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-1ef33fb3ca062b95
• 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-1ef33fb3ca062b95
• 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-1ef33fb3ca062b95
• 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-1ef33fb3ca062b95

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-1ef33fb3ca062b95

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

  • premise (sequence): Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
    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-bad4de7d401fa939 root: theorem check

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

    • preserver_image reversal, shift 0 e-3d59ef840ca39edf

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
        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-bad4de7d401fa939 (expanded above)

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.