A137338: Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.

OEIS matches ?

• A137338: triangle, 58 terms agree, 10 full rows row-aligned (tabl) in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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,      1;
  2:     0,     -3,       1;
  3:     3,      6,      -6,      1;
  4:   -12,     -9,      26,    -10,       1;
  5:    45,      3,    -109,     71,     -15,     1;
  6:  -198,     81,     501,   -475,     155,   -21,     1;
  7:  1071,   -786,   -2663,   3329,   -1455,   295,   -28,   1;
  8: -6984,   6711,   16510, -25495,   13729, -3647,   511, -36,   1;
  9: 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1;

The row polynomials satisfy P(n,x) = -(n-x)*P(n-1,x) - n*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.

P[0, x_] := 1;
P[1, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[-(n - x)*P[n - 1, x] - n*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n - x)*v[i - 1] - n*v[i - 2]); 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:
   1:     1;
   2:    -1,      1;
   3:     0,     -3,       1;
   4:     3,      6,      -6,      1;
   5:   -12,     -9,      26,    -10,       1;
   6:    45,      3,    -109,     71,     -15,     1;
   7:  -198,     81,     501,   -475,     155,   -21,     1;
   8:  1071,   -786,   -2663,   3329,   -1455,   295,   -28,   1;
   9: -6984,   6711,   16510, -25495,   13729, -3647,   511, -36,   1;
  10: 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1;

The row polynomials satisfy P(n,x) = -(n-1-x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -1 + x.

P[1, x_] := 1;
P[2, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 1 - x)*P[n - 1, x] - (n - 1)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -1 + x; for(n = 3, N, my(i = n); v[i] = -(n - 1 - x)*v[i - 1] - (n - 1)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-fe1ae6008805b5f0
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-fe1ae6008805b5f0

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 three_term_recurrence e-fe1ae6008805b5f0 root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 3 proof steps

• e-5aedd40a059ef120 preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of…
• e-6aed627db0c42e8b preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients…
• e-c9e97b9a66627f15 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the C…

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023