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
A349318
G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 2 * x).
Original entry on oeis.org
1, 1, 5, 28, 171, 1113, 7590, 53588, 388519, 2876003, 21648065, 165193576, 1275043280, 9936953788, 78087083456, 618049278976, 4922606097263, 39425205882007, 317316076325015, 2565216211152700, 20819872339143179, 169586043613302169, 1385856599443533442
Offset: 0
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nmax = 22; A[] = 0; Do[A[x] = 1 + x A[x]^3/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
A349531
G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 3 * x).
Original entry on oeis.org
1, 1, 6, 39, 271, 1986, 15171, 119694, 968589, 7997970, 67132164, 571138362, 4914229293, 42690269053, 373915274505, 3298492524831, 29279961769422, 261348675838758, 2344226713167048, 21119556517672650, 191022983002755162, 1733959471178342088, 15790787266617518790
Offset: 0
-
nmax = 22; A[] = 0; Do[A[x] = 1 + x A[x]^3/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 22}]
A349532
G.f. A(x) satisfies: A(x) = 1 + x * A(x)^3 / (1 - 4 * x).
Original entry on oeis.org
1, 1, 7, 52, 407, 3329, 28232, 246552, 2204895, 20103027, 186223399, 1748009560, 16591329652, 158975004204, 1535725632552, 14940742412112, 146259921123407, 1439658075118967, 14240062489572485, 141469058343614452, 1410975387252602527, 14122900638031585153
Offset: 0
-
nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x]^3/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = 4 a[n - 1] + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n - 1, k - 1] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]
A371889
G.f. A(x) satisfies A(x) = 1 - x/A(x)^2 * (1 - A(x) - A(x)^3).
Original entry on oeis.org
1, 1, 2, 2, -1, -4, 7, 33, -5, -200, -151, 1185, 2202, -6069, -21799, 21791, 182718, 26520, -1349611, -1613331, 8674338, 21651795, -44750412, -217666394, 121538304, 1859974399, 1023915107, -13828122997, -23155237537, 86925632115, 282182920662
Offset: 0
-
a(n) = if(n==0, 1, sum(k=0, n, binomial(n, k)*binomial(n-3*k, n-k-1))/n);
A370695
G.f. A(x) satisfies A(x) = (1 + x*A(x)^(3/4) / (1-x))^4.
Original entry on oeis.org
1, 4, 22, 128, 777, 4872, 31330, 205560, 1370868, 9266104, 63343006, 437183260, 3042337215, 21323543252, 150395596016, 1066637271424, 7602188660799, 54422262148632, 391146728466980, 2821396586367568, 20417766975784066, 148200184917042112
Offset: 0
-
A370695 := proc(n)
4*add(binomial(n-1,n-k)*binomial(3*k+4,k)/(3*k+4),k=0..n) ;
end proc:
seq(A370695(n),n=0..80) ; #R. J. Mathar, Oct 24 2024
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a(n) = 4*sum(k=0, n, binomial(n-1, n-k)*binomial(3*k+4, k)/(3*k+4));
A379192
G.f. A(x) satisfies A(x) = (1 + x*A(x))^3/(1 - x*A(x)^2).
Original entry on oeis.org
1, 4, 27, 235, 2344, 25374, 289906, 3441015, 42017262, 524418639, 6660297019, 85796763321, 1118314903447, 14722203914653, 195465862293738, 2614323606027841, 35191188308646852, 476390139438508209, 6481416282265645008, 88577523301166187997, 1215421503348039618483
Offset: 0
-
a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(3*n+3*k+3, n-k)/(n+2*k+1));
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