A376112
a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} (3^k-1) * a(k-1) * a(n-k).
Original entry on oeis.org
1, 1, 5, 74, 3119, 384099, 140605620, 153966205482, 505318125737963, 4973847408741044519, 146857822147450491641165, 13007931631590001724722114996, 3456493610037973055076316970551876, 2755388815749774181719259556096183210356, 6589473777446361501832833785593366614276353520
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/2) Sum[(3^k - 1) a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 14}]
nmax = 14; A[] = 0; Do[A[x] = 2/(2 + x A[x] - 3 x A[3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A349286
G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(3*x)^2).
Original entry on oeis.org
1, 1, 7, 148, 8695, 1455433, 715092412, 1046279830240, 4581706851060703, 60144130721142806611, 2367934845363738749478583, 279660212608573899258160179472, 99083333806943049370375436233999876, 105314387436569180750449877597844106742044
Offset: 0
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nmax = 13; A[] = 0; Do[A[x] = 1/(1 - x A[3 x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[3^(i + j) a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 13}]
A385618
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(2*x) + A(3*x)) ).
Original entry on oeis.org
1, 2, 14, 230, 9014, 913334, 254986934, 203241812630, 471322195238102, 3214892041613961206, 64937611960188470964662, 3901256965326759127330935830, 699101347969640933511109922382422, 374397435055450676411068538643233721206, 599979003238812649083869782544110463986119734
Offset: 0
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terms = 15; A[] = 1; Do[A[x] = 1/( 1 - x*(A[2*x] + A[3*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+3^j)*v[j+1]*v[i-j])); v;