A357549 - OEIS

%I #13 Dec 04 2022 13:24:00

%S 1,1,3,5,9,17,30,52,91,161,285,503,889,1573,2782,4920,8697,15368,

%T 27146,47928,84590,149246,263247,464214,818445,1442762,2543025,

%U 4482001,7898979,13920609,24532535,43234510,76195273,134288583,236682848,417170144,735325596,1296184444

%N a(n) = floor( Sum_{k=0..n-1} n^k / (k! * a(k)) ), for n > 0 with a(0) = 1.

%C Limit_{n->oo} a(n)/a(n+1) = w = exp(-w) = LambertW(1), the omega constant A030178.

%H Paul D. Hanna, <a href="/A357549/b357549.txt">Table of n, a(n) for n = 0..1000</a>

%e a(n) = floor(1 + n/a(1) + n^2/(2!*a(2)) + n^3/(3!*a(3)) + n^4/(4!*a(4)) + n^5/(5!*a(5)) + ... + n^(n-1)/((n-1)!*a(n-1)) ), for n > 0 with a(0) = 1.

%e To generate this sequence, start with a(0) = 1 and proceed as follows:

%e a(1) = 1;

%e a(2) = 1 + 2;

%e a(3) = floor(1 + 3 + 3^2/(2!*3)) = 5;

%e a(4) = floor(1 + 4 + 4^2/(2!*3) + 4^3/(3!*5)) = 9;

%e a(5) = floor(1 + 5 + 5^2/(2!*3) + 5^3/(3!*5) + 5^4/(4!*9)) = 17;

%e a(6) = floor(1 + 6 + 6^2/(2!*3) + 6^3/(3!*5) + 6^4/(4!*9) + 6^5/(5!*17)) = 30;

%e a(7) = floor(1 + 7 + 7^2/(2!*3) + 7^3/(3!*5) + 7^4/(4!*9) + 7^5/(5!*17) + 7^6/(6!*30)) = 52;

%e a(8) = floor(1 + 8 + 8^2/(2!*3) + 8^3/(3!*5) + 8^4/(4!*9) + 8^5/(5!*17) + 8^6/(6!*30) + 8^7/(7!*52)) = 91;

%e ...

%e The terms of this sequence are computed from partial sums; the actual infinite sums: Sum_{k>=0} n^k / (k!*a(k)), for n >= 1, begin:

%e n = 1: 2.205170228313619257204573175905229637440183827382...

%e n = 2: 4.026624683096007253196633437972996492234406960420...

%e n = 3: 6.938404847258827610039050722524656436473915836809...

%e n = 4: 11.76290965545838695557108226269004580813840600527...

%e n = 5: 19.93268682960501544009268973006846510258954225008...

%e n = 6: 33.95355685301572322838214122801051011301028947272...

%e n = 7: 58.22316762392820953863455561301453509123241732275...

%e n = 8: 100.4764040611128933206396099594217599817997316217...

%e n = 9: 174.3356399991557294349025383486302219269780824259...

%e n = 10: 303.8074912728852469034815183896362125031997652232...

%e ...

%o (PARI) /* Print a(n) for n = 0 through N */

%o N = 40; A=vector(N+1);

%o { a(n) = if(n<0,0, A[n+1] = if(n<1,1, floor( sum(k=0,n-1, n^k/k!/A[k+1]) ) )) }

%o for(n=0,N,print1(a(n),", "))

%Y Cf. A030178.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 01 2022