A385527
E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).
Original entry on oeis.org
1, 1, 9, 457, 118961, 152894961, 940318147705, 26967408304580857, 3534888068831469959649, 2084993641133372935803249505, 5465706581663919414225671125834601, 63043356313898446097762231466174924913065, 3173076775252515207774429654590479617164788572049
Offset: 0
-
nmax = 15; A[] = 1; Do[A[x] = E^(x*A[4*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
ary = [1]
(1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
ary
end
def A385527(n)
A(4, n)
end
A385528
E.g.f. A(x) satisfies A(x) = exp(x*A(-2*x)).
Original entry on oeis.org
1, 1, -3, -47, 1385, 119601, -22345691, -10181013695, 10346973518097, 23934447308323873, -122307331801326167539, -1379021793666951568998159, 33874331587448813081748999673, 1804181313330860398948564389193681, -206892703326367302570264123699846971211
Offset: 0
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
ary = [1]
(1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
ary
end
def A385528(n)
A(-2, n)
end
A385529
E.g.f. A(x) satisfies A(x) = exp(x*A(-3*x)).
Original entry on oeis.org
1, 1, -5, -152, 15949, 6548656, -9510189137, -48598095401792, 849885323784381337, 50192539805114962349824, -9878895951508580401879879229, -6416836884643090722807370469927936, 13640603845766595275775514993987722683941, 94239467260528503337471761892783659993298198528
Offset: 0
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
ary = [1]
(1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
ary
end
def A385529(n)
A(-3, n)
end
A385530
E.g.f. A(x) satisfies A(x) = exp(x*A(-4*x)).
Original entry on oeis.org
1, 1, -7, -359, 90705, 116586321, -715618113143, -20523234900205911, 2689857437569063003169, 1586566688643256394542888225, -4159073515698730238218108546470759, -47972197129364591236078587520718376138951, 2414519164037893898620724924577882591112859773297
Offset: 0
-
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
ary = [1]
(1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
ary
end
def A385530(n)
A(-4, n)
end
A385620
E.g.f. A(x) satisfies A(x) = exp( x*(A(2*x) + A(3*x)) ).
Original entry on oeis.org
1, 2, 24, 1064, 158144, 78427712, 130391102464, 725657074158592, 13450842239318679552, 825492067428121929359360, 166724642619378284453845213184, 110175812687250637947409895640473600, 236918101449618886434191300434062010777600, 1649425480856495624442166311045759714226010423296
Offset: 0
-
terms = 14; A[] = 1; Do[A[x] =Exp[x*(A[2*x] + A[3*x])]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 05 2025 *)
-
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+3^j)*binomial(i-1, j)*v[j+1]*v[i-j])); v;
Showing 1-5 of 5 results.