A385619
E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(2*x)) ).
Original entry on oeis.org
1, 2, 16, 320, 14176, 1363872, 288285760, 135499302976, 142083696478720, 331241746024775168, 1705949708332396248064, 19272264281263882812337152, 474329882865823082358501265408, 25275628582523724268037232839274496, 2899873213836728319564120809900380069888
Offset: 0
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terms = 15; A[] = 1; Do[A[x] =Exp[x*(A[x] + A[2*x])]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* 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, (j+1)*(2^j+1)*binomial(i-1, j)*v[j+1]*v[i-j])); v;
A385622
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(3*x))/2 ).
Original entry on oeis.org
1, 1, 3, 20, 321, 13847, 1718124, 630600310, 691143519765, 2269026118814651, 22336295204505116859, 659523795328845920952570, 58417979762116119140729740620, 15523000838307934869469597031994180, 12374377440444177691000805646758968904928, 29593162781962095695448333383964939013238970030
Offset: 0
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terms = 16; A[] = 1; Do[A[x] = 1/( 1 - x*(A[x] + A[3*x])/2 )+ 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, (3^j+1)*v[j+1]*v[i-j])/2); v;
A385648
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x))^2 ).
Original entry on oeis.org
1, 4, 64, 1872, 91328, 7563648, 1115422976, 306988895488, 162926170881024, 169827391985854464, 350891899856754294784, 1443597302250006622052352, 11851990053153536620868173824, 194396568906445310993071164686336, 6373487768490075927307409156798611456
Offset: 0
A385649
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x))^3 ).
Original entry on oeis.org
1, 8, 352, 29696, 4263424, 1049470976, 462206058496, 380751228633088, 605491779706159104, 1892234112450731442176, 11725274627114715154743296, 144692808471111027067403108352, 3563512028948515548768609167736832, 175339259291213196115801459160952864768
Offset: 0
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;
A385985
a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k + 1) * binomial(n-1,k) * a(k) * a(n-1-k).
Original entry on oeis.org
1, 2, 10, 94, 1514, 40790, 1862050, 148965310, 21742138970, 5994070800710, 3197362825740850, 3348408098259631150, 6941708283693589284650, 28621208382355252313372150, 235296246090820978083474438850, 3862393961855717768080204865278750, 126690441172711092666839985418516720250
Offset: 0
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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+1)*binomial(i-1, j)*v[j+1]*v[i-j])); v;
Showing 1-6 of 6 results.