A378326
a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(n*k,k) / ((n-1)*k+1).
Original entry on oeis.org
1, 1, 3, 19, 219, 3901, 95838, 3022909, 116798643, 5350403737, 283728025998, 17104314563843, 1155635807408096, 86513627563199279, 7109252862969177287, 636268582522962837475, 61610670571434193189443, 6418044336586421956746033, 715718717341021991299583730
Offset: 0
-
Table[Sum[Binomial[n-1, k-1]*Binomial[n*k, k]/((n-1)*k+1), {k, 0, n}], {n, 0, 20}]
A365194
G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^6).
Original entry on oeis.org
1, 1, 6, 52, 529, 5889, 69462, 853013, 10791018, 139659604, 1840435530, 24611295075, 333132371248, 4555465710569, 62839303262352, 873363902976309, 12218178082489873, 171918448407833112, 2431415226089290680, 34544425914499450493, 492807213597429920649
Offset: 0
-
a(n) = sum(k=0, n, binomial(6*n-k+1, k)*binomial(n-1, n-k)/(6*n-k+1));
A378325
G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.
Original entry on oeis.org
1, 1, 2, 7, 41, 338, 3499, 42969, 606351, 9633640, 169888025, 3290314970, 69409429043, 1584105116525, 38894316619948, 1022411500472240, 28653072049382809, 852911635849385778, 26876978490909421289, 893929164892155754432, 31296785296935394097351, 1150551256823546563078988
Offset: 0
-
{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^k, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ after Paul D. Hanna
A371583
G.f. satisfies A(x) = ( 1 + x*A(x)^(5/2) / (1 - x) )^2.
Original entry on oeis.org
1, 2, 13, 104, 940, 9166, 94044, 1000602, 10939780, 122161128, 1387361151, 15974899766, 186069556707, 2188416960148, 25953579753464, 310022550197360, 3726709235290628, 45047517497268968, 547217895030263028, 6676784544374859088, 81789906534091716353
Offset: 0
-
a(n, r=2, s=1, t=5, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));
A371913
G.f. A(x) satisfies A(x) = 1 - x/A(x)^4 * (1 - A(x) - A(x)^5).
Original entry on oeis.org
1, 1, 2, 0, -6, 12, 67, -152, -740, 2296, 9017, -35979, -113936, 579516, 1454975, -9493390, -18317155, 157178640, 220172289, -2618995381, -2377680689, 43783556265, 19149194005, -732638868460, 16196837316, 12246524817736, -5891297294673
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(n-5*k, n-k-1))/n);
A371914
G.f. A(x) satisfies A(x) = 1 - x/A(x)^3 * (1 - A(x) - A(x)^5).
Original entry on oeis.org
1, 1, 3, 7, 15, 43, 168, 626, 2005, 6245, 22266, 87365, 328727, 1154975, 4086410, 15464587, 60368094, 229327457, 847539610, 3174058754, 12277874065, 47912709420, 184171945435, 701491726600, 2700878181660, 10556457650417, 41330116314628
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(2*n-5*k, n-k-1))/n);
A371915
G.f. A(x) satisfies A(x) = 1 - x/A(x)^2 * (1 - A(x) - A(x)^5).
Original entry on oeis.org
1, 1, 4, 17, 80, 414, 2289, 13199, 78306, 474630, 2926744, 18304543, 115837726, 740379722, 4772461321, 30989448116, 202518745795, 1330961476358, 8791022012712, 58325109518331, 388523983047285, 2597516226459845, 17423367396517210, 117223205014488833
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(3*n-5*k, n-k-1))/n);