A309509
G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).
Original entry on oeis.org
0, 1, 2, 2, 2, 2, 0, 4, 6, -58, 100, 1052, -5924, -21972, 322020, 332392, -21168682, 29068598, 1724404180, -7070346036, -172304798980, 1290100381724, 20728501384592, -247269172883976, -2936888518668676, 53037176259027580, 477640220538178184
Offset: 0
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half[q_] := half[q] = Module[{h}, h[0] = 0; h[1] = 1; h[n_Integer] := h[n] = Module[{c}, c[m_Integer /; m < n] := h[m]; c[n] /. Solve[q[n] == Sum[k! c[k] BellY[n, k, Table[m! c[m], {m, n - k + 1}]], {k, n}]/n!, c[n]][[1]]]; h]; a[n_Integer] := a[n] = half[Function[k, k 2^(k-1)]][n]; Table[a[n], {n, 0, 26}]
A030274
Numerators of sequence {b(1), b(2), ...} which when COMPOSED with itself gives {1,2,3,...}.
Original entry on oeis.org
1, 1, 1, 1, 1, 0, 1, 3, -29, 25, 263, -1481, -5493, 80505, 41549, -10584341, 14534299, 431101045, -1767586509, -43076199745, 322525095431, 1295531336537, -30908646610497, -734222129667169, 13259294064756895, 59705027567272273, -1617292893727823431, -1346735121534484263
Offset: 1
1, 1, 1/2, 1/4, 1/8, 0, 1/16, 3/64, -29/128, 25/128, 263/256, -1481/512, -5493/1024, 80505/2048, ... = A030274/A030275
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t[n_, m_] := t[n, m] = If[ n == m , 1 , 1/2*(Binomial[n+m-1, 2*m-1] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}])]; a[n_] := t[n, 1] // Numerator; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
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T(n, m):=if n=m then 1 else 1/2*(binomial(n+m-1, 2*m-1)-sum(T(n, i)*T(i, m), i, m+1, n-1));
makelist(num(T(n, 1)), n, 1, 10); /* Vladimir Kruchinin, Mar 14 2012 */
A091138
E.g.f. A(x) satisfies A(A(x)) = x/(1-x)^2.
Original entry on oeis.org
1, 2, 3, 6, 15, 0, 315, 1890, -82215, 708750, 41008275, -1385549550, -33403344975, 3426898600125, 26529571443375, -13516476003780750, 157765729690193625, 84230651703487038750, -3280917943856839411125, -799561865724400084556250, 62859004972802312944044375
Offset: 1
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t[n_, m_] := t[n, m] = If[n == m, 1, 1/2*(Binomial[n+m-1, 2*m-1] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}])]; a[n_] := n!*t[n, 1]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
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T(n,m):=if n=m then 1 else 1/2*(binomial(n+m-1,2*m-1)-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(2^(n-1)*T(n,1),n,1,10); /* Vladimir Kruchinin, Mar 14 2012 */
A334284
Denominators of power series coefficients of A(x) satisfying A(A(x)) = x + Sum_{k>=2} prime(k-1) * x^k.
Original entry on oeis.org
1, 1, 2, 4, 8, 8, 16, 64, 128, 64, 256, 512, 1024, 2048, 2048, 16384, 32768, 16384, 65536, 131072, 262144, 262144, 524288, 2097152, 4194304, 2097152, 8388608, 8388608, 16777216, 134217728, 67108864, 1073741824, 2147483648, 1073741824, 4294967296
Offset: 1
1, 1, 1/2, 3/4, -3/8, 11/8, -47/16, 291/64, -361/128, -327/64, 2651/256, 8117/512, -23761/1024, -920509/2048, ...
Showing 1-4 of 4 results.
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