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

Author

Peter Luschny, May 23 2017

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

• Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.

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.

Programs

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}]

Formula

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