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%I #16 Mar 21 2020 12:04:42
%S 1,6,29,120,436,1484,4841,15352,47695,145855,440529,1317230,3906114,
%T 11502747,33672919,98070520,284355536,821268392,2363758888,6782327435,
%U 19406607815,55390260847,157736165229,448260958526,1271477862231,3600244966868,10177939690298,28730604992496,80990395600321,228017389234353,641188474891466,1801028679245339,5053629451691563,14166476265870459,39675398491866930
%N a(n) = floor( Sum_{k>=0} n^sqrt(k) / Gamma(sqrt(k) + 1) ), where Gamma is Euler's gamma function.
%F a(n) = floor( 1 + n + n^sqrt(2)/Gamma(sqrt(2)+1) + n^sqrt(3)/Gamma(sqrt(3)+1) + n^2/2! + n^sqrt(5)/Gamma(sqrt(5)+1) + n^sqrt(6)/Gamma(sqrt(6)+1) + n^sqrt(7)/Gamma(sqrt(7)+1) + n^sqrt(8)/Gamma(sqrt(8)+1) + n^3/3! + ... ).
%F Conjecture: a(n) ~ 2*n*exp(n). - _Vaclav Kotesovec_, Sep 16 2019
%e Sample of actual sums:
%e n | Sum_{k>=0} n^sqrt(k) / gamma(sqrt(k) + 1)
%e ---+------------------------------------------
%e 0 | 1
%e 1 | 6.0508649446787330759292180438672944450...
%e 2 | 29.968525272075391841774353716455445560...
%e 3 | 120.77771764573995394225247006780774786...
%e 4 | 436.93096230917013502156544902769718883...
%e 5 | 1484.1772399595796976511254713998475451...
%e 6 | 4841.1041818159327351259845350253722329...
%e 7 | 15352.745719315385435796595860510779971...
%e 8 | 47695.139818009834449468837174136343367...
%e 9 | 145855.25944112199314551854304392768195...
%e 10 | 440529.00647863443505456127264544798356...
%e 11 | 1317230.7544650817155211352616976188752...
%e 12 | 3906114.5806224559822936126739714260866...
%e 13 | 11502747.731829540330662657985445881916...
%e 14 | 33672919.452042528560668451915846157284...
%e 15 | 98070520.627899598703112706179080116295...
%e 16 | 284355536.07050365983139838540777869579...
%e 17 | 821268392.99830998622537351585444700214...
%e 18 | 2363758888.2849378475946066480749874366...
%e 19 | 6782327435.9080727036686878106620171299...
%e 20 | 19406607815.665971565137700836799078913...
%o (PARI) {a(n) = if(n==0,1, floor( suminf(k=0, n^sqrt(k) / gamma(sqrt(k) + 1) ) ) )}
%o for(n=0,40,print1(a(n),", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 14 2019