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%I #23 Jan 29 2023 23:01:08
%S 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 252,0,1,12,270,5120,44730,63504,924,0,1,14,396,10900,190120,1172556,
%U 853776,3432,0,1,16,546,19920,551950,7939008,32496156,11778624,12870,0
%N Square array A(n,k) = (2*n)! [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.
%F A(n,k) = A287316(n,k) * binomial(2*n,n).
%e Arrays start:
%e k\n| 0 1 2 3 4 5 6
%e ---|---------------------------------------------------------
%e k=0| 1, 0, 0, 0, 0, 0, 0, ... A000007
%e k=1| 1, 2, 6, 20, 70, 252, 924, ... A000984
%e k=2| 1, 4, 36, 400, 4900, 63504, 853776, ... A002894
%e k=3| 1, 6, 90, 1860, 44730, 1172556, 32496156, ... A002896
%e k=4| 1, 8, 168, 5120, 190120, 7939008, 357713664, ... A039699
%e k=5| 1, 10, 270, 10900, 551950, 32232060, 2070891900, ... A287317
%e k=6| 1, 12, 396, 19920, 1281420, 96807312, 8175770064, ... A356258
%e k=7| 1, 14, 546, 32900, 2570050, 238935564, 25142196156, ...
%e k=8| 1, 16, 720, 50560, 4649680, 514031616, 64941883776, ...
%e k=9| 1, 18, 918, 73620, 7792470, 999283068, 147563170524, ...
%p A287318_row := proc(k, len) local b, ser;
%p b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len);
%p seq((2*i)!*coeff(ser,x,i), i=0..len-1) end:
%p for k from 0 to 6 do A287318_row(k, 9) od;
%t Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (2 n)!, {n, 0, 6}], {k, 0, 6}]
%Y Rows: A000007 (k=0), A000984 (k=1), A002894 (k=2), A002896 (k=3), A039699 (k=4), A287317 (k=5), A356258 (k=6).
%Y Columns: A005843 (n=1), A152746 (n=2), 20*A169711 (n=3), 70*A169712 (n=4), 252*A169713 (n=5).
%Y Main diagonal gives A303503.
%Y Cf. A287316.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, May 23 2017