A367919
Expansion of e.g.f. exp(4*(exp(x) - 1) - x).
Original entry on oeis.org
1, 3, 13, 67, 397, 2627, 19085, 150339, 1272205, 11481155, 109852813, 1109011779, 11765211021, 130707706435, 1516160466573, 18314760232771, 229865470694797, 2991427959247939, 40292570823959693, 560791503840522563, 8053114165521427341, 119158887402348541507
Offset: 0
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nmax = 21; CoefficientList[Series[Exp[4 (Exp[x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A367920
Expansion of e.g.f. exp(4*(exp(x) - 1) - 2*x).
Original entry on oeis.org
1, 2, 8, 36, 196, 1196, 8116, 60108, 481140, 4126540, 37671540, 364068172, 3707910772, 39645022540, 443540780660, 5177560304972, 62903920321140, 793654042136908, 10378403752717940, 140413475790402892, 1962339063781284468, 28287778534523140428, 420059992540347885172
Offset: 0
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nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
A367921
Expansion of e.g.f. exp(4*(exp(x) - 1) - 3*x).
Original entry on oeis.org
1, 1, 5, 17, 93, 505, 3269, 22657, 172461, 1407177, 12284629, 113832273, 1114775869, 11487315481, 124118143717, 1401808691489, 16504815145421, 202101235848297, 2568312461002741, 33808677627863537, 460227870278020957, 6468672644291075001, 93745096205219336709
Offset: 0
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b:= proc(n, k, m) option remember; `if`(n=0, 4^m, `if`(k>0,
b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22); # Alois P. Heinz, Apr 29 2025
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nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1) - 3 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
A375872
E.g.f. satisfies A(x) = exp( 4 * (exp(x*A(x)^(3/4)) - 1) ).
Original entry on oeis.org
1, 4, 44, 836, 22980, 832888, 37615340, 2038258804, 128989219860, 9343113460308, 762596057754748, 69273095355130488, 6932765720797549924, 758009268677055714964, 89907747171907593677068, 11498798927333436173636612, 1577528093912610651931113908
Offset: 0
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a(n) = 4*sum(k=0, n, (3*n+4)^(k-1)*stirling(n, k, 2));