A380516
Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689
Offset: 0
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Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *)
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a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));
A380511
Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
Original entry on oeis.org
1, 1, 5, 55, 961, 23141, 711421, 26631235, 1175535425, 59786520841, 3442729157461, 221413508687471, 15730688410899265, 1223574846548300845, 103417508018836074701, 9437941200860641295611, 924934291227615821904001, 96881241931552168636182545, 10801002623361396194857667365
Offset: 0
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a(n) = if(n==0, 1, 2*n!*sum(k=0, n-1, binomial(2*n+k, k)/((2*n+k)*(n-k-1)!)));
A380513
Expansion of e.g.f. exp(x*G(x)) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 3, 31, 649, 20241, 831691, 42281023, 2558247441, 179401012129, 14301145772371, 1276863732880671, 126200478678828313, 13677209933635675441, 1612657716714084149019, 205505541279096688937791, 28144314031348292162103841, 4122178445898981809990411073, 642961375302043479923591655331
Offset: 0
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a(n) = if(n==0, 1, n!*sum(k=0, n-1, binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));
A380514
Expansion of e.g.f. exp(x*G(x)^2) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 1, 5, 67, 1537, 50021, 2107021, 108885295, 6665443457, 471522589417, 37843890892021, 3397250515809371, 337267132243022785, 36687625652474612557, 4339368321317331858557, 554467482301151809302151, 76112537023512618262963201, 11170667360636927554290623825, 1745500813880455301486766050917
Offset: 0
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a(n) = if(n==0, 1, 2*n!*sum(k=0, n-1, binomial(2*n+2*k, k)/((2*n+2*k)*(n-k-1)!)));
A250917
Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.
Original entry on oeis.org
1, 1, 7, 73, 1033, 18541, 403831, 10351237, 305355793, 10192132153, 379819484551, 15634219476481, 704566985120857, 34506514429777573, 1825081888365736183, 103685565729559782781, 6297505655719537293601, 407233553972252986277617, 27935786938445348562454663
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
such that log(A(x)) = x*C(x)^3,
log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
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{a(n)=my(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
n!*polcoef(exp(x*C^3), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n) = if(n==0, 1, sum(k=0, n, n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) ))}
for(n=0, 20, print1(a(n), ", "))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^3/x^2))) \\ Seiichi Manyama, Mar 15 2025
A380605
Expansion of e.g.f. exp(2*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 2, 16, 260, 6544, 224672, 9797824, 518778752, 32332764160, 2319086302208, 188178044545024, 17043816700333568, 1704575787500099584, 186577340672207974400, 22185432394552519868416, 2847773562263558405439488, 392481896442656581445287936, 57805399208817471918851883008
Offset: 0
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a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, 2^(n-k)*binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));
A380606
Expansion of e.g.f. exp(3*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
Original entry on oeis.org
1, 3, 27, 459, 11817, 411183, 18090459, 963856071, 60351513777, 4344290172891, 353515902334299, 32093341598006307, 3215888732193019353, 352572962113533923271, 41981774097966848444763, 5395346708265250105968927, 744369113570455426540767201, 109733083289828610273889269939
Offset: 0
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a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, 3^(n-k)*binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));
Showing 1-7 of 7 results.
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