A101483 Column 2 of triangular matrix T=A101479, in which row n equals row (n-1) of T^(n-1) followed by '1'.
1, 1, 3, 18, 170, 2220, 37149, 758814, 18301950, 508907970, 16023271660, 563336380167, 21870097514265, 929083211885920, 42864992943282825, 2134180938818369850, 114045356939250133338, 6510275427522198876426
Offset: 0
Keywords
Examples
This sequence can also be generated in the following manner. Start a table with the all 1's sequence in row 0; from then on, row n+1 can be formed from row n by dropping the initial n+1 terms of row n and taking partial sums of the remaining terms to obtain row n+1. The table below illustrates this method: [1], 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...; [1, 2], 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...; [3, 7, 12], 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, ...; [18, 43, 76, 118], 170, 233, 308, 396, 498, 615, 748, 898, ...; [170, 403, 711, 1107, 1605], 2220, 2968, 3866, 4932, 6185, ...; [2220, 5188, 9054, 13986, 20171, 27816], 37149, 48420, ...; [37149, 85569, 147471, 225363, 322075, 440785, 585046], 758814, ...; ... In the above table, drop the initial n+1 terms in row n (enclosed in square brackets) and then take partial sums to obtain row n+1 for n>=0; this sequence then forms the first column of the resultant table. Note: column k of the above table equals column 2 of matrix power T^(k+1) where T=A101479, for k>=0.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..370
Programs
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PARI
{a(n)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=(A^i)[i-1,j]);));A=B);return(A[n+1,1])} for(n=0,25,print1(a(n),", ")) -
PARI
{a(n)=local(A=[1]);for(i=1,n,A=Vec(Ser(A)/(1-x)^(#A));A=concat(A,A[#A]));A[#A]} for(n=0,25,print1(a(n),", "))
Formula
a(n) ~ (2 + LambertW(-2*exp(-2))) * exp(n+2) * n^(n - 3/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Dec 03 2017