A287318 Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 36, 20, 0, 1, 8, 90, 400, 70, 0, 1, 10, 168, 1860, 4900, 252, 0, 1, 12, 270, 5120, 44730, 63504, 924, 0, 1, 14, 396, 10900, 190120, 1172556, 853776, 3432, 0, 1, 16, 546, 19920, 551950, 7939008, 32496156, 11778624, 12870, 0

Offset

0,5

Links

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

Formula

A(n,k) = A287316(n,k) * binomial(2*n,n).

Example

Arrays start:
k\n| 0   1    2      3        4          5           6
---|---------------------------------------------------------
k=0| 1,  0,   0,     0,       0,         0,            0, ... A000007
k=1| 1,  2,   6,    20,      70,       252,          924, ... A000984
k=2| 1,  4,  36,   400,    4900,     63504,       853776, ... A002894
k=3| 1,  6,  90,  1860,   44730,   1172556,     32496156, ... A002896
k=4| 1,  8, 168,  5120,  190120,   7939008,    357713664, ... A039699
k=5| 1, 10, 270, 10900,  551950,  32232060,   2070891900, ... A287317
k=6| 1, 12, 396, 19920, 1281420,  96807312,   8175770064, ... A356258
k=7| 1, 14, 546, 32900, 2570050, 238935564,  25142196156, ...
k=8| 1, 16, 720, 50560, 4649680, 514031616,  64941883776, ...
k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...

Maple

A287318_row := proc(k, len) local b, ser;
b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
seq((2*i)!*coeff(ser, x, i), i=0..len-1) end:
for k from 0 to 6 do A287318_row(k, 9) od;

Mathematica

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]

Crossrefs

Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).

Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).

Main diagonal gives A303503.

Cf. A287316.

Sequence in context: A128749 A106579 A378318 * A329020 A351640 A173003

Adjacent sequences: A287315 A287316 A287317 * A287319 A287320 A287321

Keyword

nonn,tabl

Author

Peter Luschny, May 23 2017

Status

approved