A062264 → x ↦ x+1

OEIS matches ?

• A386882: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:      6,       5;
  2:     28,      42,      15;
  3:    120,     252,     168,      35;
  4:    495,    1320,    1260,     504,      70;
  5:   2002,    6435,    7920,    4620,    1260,     126;
  6:   8008,   30030,   45045,   34320,   13860,    2772,    210;
  7:  31824,  136136,  240240,  225225,  120120,   36036,   5544,    330;
  8: 125970,  604656, 1225224, 1361360,  900900,  360360,  84084,  10296,   495;
  9: 497420, 2645370, 6046560, 7759752, 6126120, 3063060, 960960, 180180, 18018, 715;

The row polynomials satisfy (n^2+4*n)*P(n,x) = (4*n^2+14*n+12+(3*n^2+10*n+12)*x)*P(n-1,x) - ((2*n+4)*x+(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (3*n^2 + 10*n + 12)*x)*P[n - 1, x] - ((2*n + 4)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 14*n + 12 + (3*n^2 + 10*n + 12)*x)*v[i - 1] - ((2*n + 4)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-06bdf5b2aa1ff3d5
• 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-06bdf5b2aa1ff3d5
• 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-06bdf5b2aa1ff3d5
• 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-06bdf5b2aa1ff3d5

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 affine, shift 0 e-06bdf5b2aa1ff3d5

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Coefficient triangle of certain polynomials N(4; m,x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-c22f7d753efa18e6 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [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