A296544
Expansion of e.g.f. sech(exp(x)-1).
Original entry on oeis.org
1, 0, -1, -3, -2, 35, 233, 406, -6463, -72951, -234204, 3454165, 59129541, 299098488, -3983040037, -98320221991, -717243756890, 8306677424447, 290745526287213, 2888789037885782, -27716446593438731, -1389236305462658939, -18071746125863480076, 133495439954224143065, 10011730596035533451801
Offset: 0
sech(exp(x)-1) = 1 - x^2/2! - 3*x^3/3! - 2*x^4/4! + 35*x^5/5! + 233*x^6/6! + ...
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a:=series(sech(exp(x)-1),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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nmax = 24; CoefficientList[Series[Sech[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
A384836
a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n,4*k)|.
Original entry on oeis.org
1, 0, 0, 0, 1, 10, 85, 735, 6770, 67320, 724550, 8427650, 105615500, 1420941600, 20448793300, 313670857500, 5111631733000, 88224807112000, 1608190674259000, 30879323250633000, 623074177992110000, 13182400475167560000, 291842125111122170000, 6748135840840046510000
Offset: 0
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Table[Sum[Abs[StirlingS1[n, 4*k]], {k, 0, Floor[n/4]}], {n, 0, 30}]
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a(n) = sum(k=0, n\4, abs(stirling(n, 4*k, 1))); \\ Michel Marcus, Jun 10 2025
A384837
a(n) = Sum_{k=0..floor(n/5)} |Stirling1(n,5*k)|.
Original entry on oeis.org
1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269326, 3416985, 45997655, 657262606, 9959178229, 159758917956, 2707741441460, 48389066401764, 909877831207125, 17965423056654249, 371766710374672096, 8047954162682335066, 181941000229690525197, 4288430328840863166236, 105226297616943093770399
Offset: 0
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Table[Sum[Abs[StirlingS1[n, 5*k]], {k, 0, Floor[n/5]}], {n, 0, 30}]
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a(n) = sum(k=0, n\5, abs(stirling(n, 5*k, 1))); \\ Michel Marcus, Jun 10 2025
A296543
Expansion of e.g.f. tanh(exp(x)-1).
Original entry on oeis.org
0, 1, 1, -1, -11, -33, 61, 1367, 7253, -12561, -580499, -4701497, 4669765, 580325215, 6636339165, 1365901495, -1122870368715, -17289945450289, -31110588453299, 3713822629274023, 74717183313957413, 280555705771423039, -19253195126787261507, -496715617694137066089, -3008746115751273626347
Offset: 0
tanh(exp(x)-1) = x/1! + x^2/2! - x^3/3! - 11*x^4/4! - 33*x^5/5! + 61*x^6/6! + 1367*x^7/7! + ...
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a:=series(tanh(exp(x)-1),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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nmax = 24; CoefficientList[Series[Tanh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Sinh[Exp[x] - 1]/Cosh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[(Exp[x] - 1)/(1 + ContinuedFractionK[(Exp[x] - 1)^2, 2 k - 1, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
A330041
Expansion of e.g.f. exp(cosh(exp(x) - 1) - 1).
Original entry on oeis.org
1, 0, 1, 3, 11, 55, 322, 2114, 15556, 127005, 1135374, 11011220, 115080825, 1288589757, 15379512670, 194796087841, 2608470709562, 36805935282625, 545626818921885, 8475730766054047, 137637670315066835, 2331584745107027528, 41122505417366272200
Offset: 0
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g:= proc(n) option remember; `if`(n=0, 1, add(
binomial(2*n-1, 2*k-1) *g(n-k), k=1..n))
end:
b:= proc(n, m) option remember; `if`(n=0,
`if`(m::odd, 0, g(m/2)), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..22); # Alois P. Heinz, Jun 23 2023
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nmax = 22; CoefficientList[Series[Exp[Cosh[Exp[x] - 1] - 1], {x, 0, nmax}], x] Range[0, nmax]!
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