cp's OEIS Frontend

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

%I A287318 #28 Jun 23 2025 18:33:58
%S A287318 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,
%T A287318 252,0,1,12,270,5120,44730,63504,924,0,1,14,396,10900,190120,1172556,
%U A287318 853776,3432,0,1,16,546,19920,551950,7939008,32496156,11778624,12870,0
%N A287318 Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.
%H A287318 Nikolai Beluhov, <a href="https://arxiv.org/abs/2506.12789">Powers of 2 in High-Dimensional Lattice Walks</a>, arXiv:2506.12789 [math.CO], 2025. See p. 19.
%F A287318 A(n,k) = A287316(n,k) * binomial(2*n,n).
%e A287318 Arrays start:
%e A287318   k\n| 0   1    2      3        4          5           6
%e A287318   ---|---------------------------------------------------------
%e A287318   k=0| 1,  0,   0,     0,       0,         0,            0, ... A000007
%e A287318   k=1| 1,  2,   6,    20,      70,       252,          924, ... A000984
%e A287318   k=2| 1,  4,  36,   400,    4900,     63504,       853776, ... A002894
%e A287318   k=3| 1,  6,  90,  1860,   44730,   1172556,     32496156, ... A002896
%e A287318   k=4| 1,  8, 168,  5120,  190120,   7939008,    357713664, ... A039699
%e A287318   k=5| 1, 10, 270, 10900,  551950,  32232060,   2070891900, ... A287317
%e A287318   k=6| 1, 12, 396, 19920, 1281420,  96807312,   8175770064, ... A356258
%e A287318   k=7| 1, 14, 546, 32900, 2570050, 238935564,  25142196156, ...
%e A287318   k=8| 1, 16, 720, 50560, 4649680, 514031616,  64941883776, ...
%e A287318   k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...
%p A287318 A287318_row := proc(k, len) local b, ser;
%p A287318 b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
%p A287318 seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:
%p A287318 for k from 0 to 6 do A287318_row(k, 9) od;
%t A287318 Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]
%Y A287318 Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).
%Y A287318 Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).
%Y A287318 Main diagonal gives A303503.
%Y A287318 Cf. A287316.
%K A287318 nonn,tabl
%O A287318 0,5
%A A287318 _Peter Luschny_, May 23 2017