A102365: Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

OEIS matches ?

• A156920: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A014307: 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: 1;
  2: 1,    1;
  3: 1,    5,      1;
  4: 1,   18,     15,      1;
  5: 1,   58,    129,     37,      1;
  6: 1,  179,    877,    646,     83,      1;
  7: 1,  543,   5280,   8030,   2685,    177,     1;
  8: 1, 1636,  29658,  82610,  56285,  10002,   367,   1;
  9: 1, 4916, 159742, 756218, 919615, 335162, 34777, 749, 1;

The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (2*x-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[(1 + (n - 1)*x)*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
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] = (1 + (n - 1)*x)*v[i - 1] + (2*x - x^2)*deriv(v[i - 1], x)); 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)

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    5,      1;
  3: 1,   15,     18,       1;
  4: 1,   37,    129,      58,       1;
  5: 1,   83,    646,     877,     179,        1;
  6: 1,  177,   2685,    8030,    5280,      543,       1;
  7: 1,  367,  10002,   56285,   82610,    29658,    1636,      1;
  8: 1,  749,  34777,  335162,  919615,   756218,  159742,   4916,     1;
  9: 1, 1515, 115566, 1792749, 8284857, 12813843, 6411720, 838038, 14757, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+n*x)*P(n-1,x) + (2*x-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[(1 + n*x)*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (2*x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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-1ce099339e80292c
• 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-1ce099339e80292c
• 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-1ce099339e80292c
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-1ce099339e80292c

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

• theorem_check liu_wang e-1ce099339e80292c root: theorem check

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

Used in 7 proof steps

• e-9fcf9816b8c5606e obreschkoff_combination → obreschkoff_combination of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3…
• e-9fcf9816b8c5606e obreschkoff_combination → obreschkoff_combination of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3…
• e-121b28b1325b14d7 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5,…
• e-2922bfe17192604c preserver_image (derivative_polynomial, shift 0) → derivative of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7,…
• e-5503999bc1e6bd41 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0…
• e-872bb58370a3a8d0 preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3,…
• e-ff253c56ca542933 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0,…

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