A100064 Triangle, read by rows, of coefficients in powers of e.g.f. for A100065 such that, for each row n>=0, Sum_{k=0..n} T(n,k)/k! = [exp(n)] (integer floor of e^n).

1, 1, 1, 1, 2, 8, 1, 3, 15, 51, 1, 4, 24, 120, 408, 1, 5, 35, 225, 1215, 4365, 1, 6, 48, 372, 2628, 15084, 53856, 1, 7, 63, 567, 4851, 36603, 216405, 777609, 1, 8, 80, 816, 8112, 74352, 585792, 3558672, 12810240, 1, 9, 99, 1125, 12663, 135081, 1301157, 10623231, 65821329, 237788055

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Example

Rows form the initial coefficients of powers of e.g.f. of A100065:
G100065^0: [1,__0,0,0,0,0,0,0,0,...],
G100065^1: [1,1,__3,-3,-57,369,3861,-76617,-413775,...],
G100065^2: [1,2,8,__12,-84,-12,7200,-40716,-1301328,...],
G100065^3: [1,3,15,51,__27,-513,4077,33237,-1211895,...],
G100065^4: [1,4,24,120,408,__216,-3168,45576,-202176,...],
G100065^5: [1,5,35,225,1215,4365,__1485,-27765,440865,...],
G100065^6: [1,6,48,372,2628,15084,53856,__10908,-282960,...],
G100065^7: [1,7,63,567,4851,36603,216405,777609,__93177,...],
G100065^8: [1,8,80,816,8112,74352,585792,3558672,12810240,...],...
such that for each row n, Sum_{k=0..n} T(n,k)/k! = [exp(n)]:
[exp(0)] = 1 = 1
[exp(1)] = 1+1 = 2
[exp(2)] = 1+2+8/2! = 7
[exp(3)] = 1+3+15/2!+51/3! = 20
[exp(4)] = 1+4+24/2!+120/3!+408/4! = 54
[exp(5)] = 1+5+35/2!+225/3!+1215/4!+4365/5! = 148

Prog

(PARI) {T(n, k)=if(n==0, 1, if(k==0, 1, if(k==n, n!*(floor(exp(n))-sum(j=0, n-1, T(n, j)/j!)), k!*polcoeff((Ser(vector(n, i, T(n-1, i-1)/(i-1)!), x)+x*O(x^k))^(n/(n-1)), k, x))))}

Crossrefs

Cf. A100065.

Sequence in context: A155694 A200689 A143198 * A352589 A275980 A343918

Adjacent sequences: A100061 A100062 A100063 * A100065 A100066 A100067

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 02 2004

Extensions

3 more terms from Michel Marcus, Jan 31 2023

Status

approved