A362885 Array read by ascending antidiagonals: A(n, k) = (1 + 2*n)*k^n.

1, 0, 1, 0, 3, 1, 0, 5, 6, 1, 0, 7, 20, 9, 1, 0, 9, 56, 45, 12, 1, 0, 11, 144, 189, 80, 15, 1, 0, 13, 352, 729, 448, 125, 18, 1, 0, 15, 832, 2673, 2304, 875, 180, 21, 1, 0, 17, 1920, 9477, 11264, 5625, 1512, 245, 24, 1, 0, 19, 4352, 32805, 53248, 34375, 11664, 2401, 320, 27, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Formula

A(n, k) = A005408(n)*A004248(n, k).

O.g.f. of column k: (1 + k*x)/(1 - k*x)^2.

E.g.f. of column k: exp(k*x)*(1 + 2*k*x).

A(n, n) = A176043(n+1).

Example

The array begins:
    1,  1,   1,    1,     1,     1, ...
    0,  3,   6,    9,    12,    15, ...
    0,  5,  20,   45,    80,   125, ...
    0,  7,  56,  189,   448,   875, ...
    0,  9, 144,  729,  2304,  5625, ...
    0, 11, 352, 2673, 11264, 34375, ...
    ...

Mathematica

A[n_, k_]:=(1+2n)k^n; Join[{1}, Table[A[n-k, k], {n, 10}, {k, 0, n}]]//Flatten (* or *)
A[n_, k_]:=SeriesCoefficient[(1+k*x)/(1-k*x)^2, {x, 0, n}]; Table[A[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten (* or *)
A[n_, k_]:=n!SeriesCoefficient[Exp[k*x](1+2k*x), {x, 0, n}]; Table[A[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten

Crossrefs

Cf. A000007 (k=0), A000012 (n=0), A004248, A005408 (k=1), A008585 (n=1), A014480 (k=2), A033429 (n=2), A058962 (k=4), A124647 (k=3), A155988 (k=9), A171220 (k=5), A176043, A199299 (k=6), A199300 (k=7), A199301 (k=8), A244727 (n=3), A362886 (antidiagonal sums).

Sequence in context: A347924 A341103 A021326 * A227342 A329989 A110032

Adjacent sequences: A362882 A362883 A362884 * A362886 A362887 A362888

Keyword

nonn,tabl

Author

Stefano Spezia, May 08 2023

Status

approved