A320253 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k - Sum_{j=1..k} exp(j*x)).

1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 23, 13, 0, 1, 10, 86, 261, 75, 0, 1, 15, 230, 1836, 3947, 541, 0, 1, 21, 505, 7900, 52250, 74613, 4683, 0, 1, 28, 973, 25425, 361754, 1858716, 1692563, 47293, 0, 1, 36, 1708, 67473, 1706629, 20706700, 79345346, 44794221, 545835, 0

Offset

0,8

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f. of column k: 1/(1 + k - exp(x)*(exp(k*x) - 1)/(exp(x) - 1)).

Example

E.g.f. of column k: A_k(x) = 1 + (1/2)*k*(k + 1)*x/1! + (1/6)*k*(3*k^3 + 8*k^2 + 6*k + 1)*x^2/2! + (1/4)*k^2*(k + 1)^2*(3*k^2 + 7*k + 3)*x^3/3! + (1/30)*k*(45*k^7 + 270*k^6 + 635*k^5 + 741*k^4 + 440*k^3 + 115*k^2 + 5*k - 1)*x^4/4! + ...
Square array begins:
  1,    1,      1,        1,         1,          1,  ...
  0,    1,      3,        6,        10,         15,  ...
  0,    3,     23,       86,       230,        505,  ...
  0,   13,    261,     1836,      7900,      25425,  ...
  0,   75,   3947,    52250,    361754,    1706629,  ...
  0,  541,  74613,  1858716,  20706700,  143195025,  ...

Mathematica

Table[Function[k, n! SeriesCoefficient[1/(1 + k - Sum[Exp[i x], {i, 1, k}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, n! SeriesCoefficient[1/(1 + k - Exp[x] (Exp[k x] - 1)/(Exp[x] - 1)), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten

Crossrefs

Columns k=0..10 give A000007, A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708.

Main diagonal gives A319508.

Sequence in context: A102752 A104548 A085707 * A141947 A216804 A010607

Adjacent sequences: A320250 A320251 A320252 * A320254 A320255 A320256

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Oct 08 2018

Status

approved