A369616
Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).
Original entry on oeis.org
1, 3, 12, 58, 314, 1824, 11107, 69955, 451918, 2977834, 19936332, 135225006, 927267595, 6417580459, 44770275705, 314489676679, 2222549047262, 15791353483602, 112734135824404, 808247711066688, 5817056710700424, 42012120642574732, 304384379305912686
Offset: 0
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A369616 := proc(n)
add(binomial(n+1,k) * binomial(3*n-3*k+1,n-k),k=0..n) ;
%/(n+1) ;
end proc;
seq(A369616(n),n=0..70) ; # R. J. Mathar, Jan 28 2024
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^2+x))/x)
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a(n) = sum(k=0, n, binomial(n+1, k)*binomial(3*n-3*k+1, n-k))/(n+1);
A379188
G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^3).
Original entry on oeis.org
1, 4, 34, 392, 5271, 77530, 1208602, 19620262, 328167191, 5616065633, 97867738285, 1730732539345, 30981439344096, 560293394484145, 10221582080782452, 187884236846039893, 3476266045318846245, 64690833375603622619, 1210026171180264742927, 22736845507710710652858
Offset: 0
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a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+3*k+1));
A370844
Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^4 + x) ).
Original entry on oeis.org
1, 5, 35, 295, 2760, 27556, 287564, 3098780, 34216020, 385106280, 4401850866, 50957904938, 596231618166, 7039674475190, 83767631913840, 1003564049999916, 12094813260406732, 146534778450346908, 1783695235540931924, 21803615393276536720, 267537602528379374851
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^4+x))/x)
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a(n) = sum(k=0, n, binomial(n, k)*binomial(5*k+3, k)/(k+1));
A379187
G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x))^3).
Original entry on oeis.org
1, 4, 30, 286, 3091, 36063, 442898, 5642628, 73893561, 988585443, 13453580815, 185661101085, 2592069904059, 36545520229810, 519601325300487, 7441580996167052, 107255985242888943, 1554576968046707916, 22644622298400113411, 331322620547205661043
Offset: 0
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a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+2*k+2, n-k)/(n+2*k+1));
Showing 1-4 of 4 results.