A326805 a(n) = floor( Sum_{k>=0} n^sqrt(k) / Gamma(sqrt(k) + 1) ), where Gamma is Euler's gamma function.

1, 6, 29, 120, 436, 1484, 4841, 15352, 47695, 145855, 440529, 1317230, 3906114, 11502747, 33672919, 98070520, 284355536, 821268392, 2363758888, 6782327435, 19406607815, 55390260847, 157736165229, 448260958526, 1271477862231, 3600244966868, 10177939690298, 28730604992496, 80990395600321, 228017389234353, 641188474891466, 1801028679245339, 5053629451691563, 14166476265870459, 39675398491866930

Offset

0,2

Links

Table of n, a(n) for n=0..34.

Formula

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! + ... ).

Conjecture: a(n) ~ 2*n*exp(n). - Vaclav Kotesovec, Sep 16 2019

Example

Sample of actual sums:
   n | Sum_{k>=0} n^sqrt(k) / gamma(sqrt(k) + 1)
  ---+------------------------------------------
   0 | 1
   1 | 6.0508649446787330759292180438672944450...
   2 | 29.968525272075391841774353716455445560...
   3 | 120.77771764573995394225247006780774786...
   4 | 436.93096230917013502156544902769718883...
   5 | 1484.1772399595796976511254713998475451...
   6 | 4841.1041818159327351259845350253722329...
   7 | 15352.745719315385435796595860510779971...
   8 | 47695.139818009834449468837174136343367...
   9 | 145855.25944112199314551854304392768195...
  10 | 440529.00647863443505456127264544798356...
  11 | 1317230.7544650817155211352616976188752...
  12 | 3906114.5806224559822936126739714260866...
  13 | 11502747.731829540330662657985445881916...
  14 | 33672919.452042528560668451915846157284...
  15 | 98070520.627899598703112706179080116295...
  16 | 284355536.07050365983139838540777869579...
  17 | 821268392.99830998622537351585444700214...
  18 | 2363758888.2849378475946066480749874366...
  19 | 6782327435.9080727036686878106620171299...
  20 | 19406607815.665971565137700836799078913...

Prog

(PARI) {a(n) = if(n==0, 1, floor( suminf(k=0, n^sqrt(k) / gamma(sqrt(k) + 1) ) ) )}
for(n=0, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A047923 A006816 A184130 * A061648 A281050 A267774

Adjacent sequences: A326802 A326803 A326804 * A326806 A326807 A326808

Keyword

nonn

Author

Paul D. Hanna, Sep 14 2019

Status

approved