A381429
E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x) * A(x)^3 ).
Original entry on oeis.org
1, 1, 8, 133, 3392, 117601, 5167808, 275334613, 17250670592, 1242994578721, 101273185092608, 9206681997173893, 923928346115182592, 101453787213382443841, 12100018549609932996608, 1557645163271323384461973, 215265839194368088629051392, 31788685348087376561935104961
Offset: 0
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a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(4*k+1, k)/(4*k+1)*a136630(n, k));
A088182
E.g.f.: 1/(1-sinh(x)-x-x^2/2).
Original entry on oeis.org
2, 9, 61, 550, 6201, 83894, 1324177, 23886552, 484745029, 10930266652, 271107065043, 7335664739264, 215030576905171, 6788059632853620, 229590704738375917, 8283076586956304128, 317510426598228001881
Offset: 1
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With[{nn=20},Rest[CoefficientList[Series[1/(1-Sinh[x]-x-x^2/2),{x,0,nn}], x]Range[0,nn]!]] (* Harvey P. Dale, May 21 2012 *)
A088189
Expansion of e.g.f.: 1/(1-sinh(x)-x-x^2).
Original entry on oeis.org
1, 2, 10, 73, 712, 8681, 127004, 2167789, 42287088, 928006801, 22628305012, 606939739253, 17759375280200, 562952693773081, 19217673118604172, 702899000556953437, 27422924828816762848, 1136746326170994466337, 49892691063347466789860, 2311486678172789188448965, 112725522031113629713962552
Offset: 0
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CoefficientList[Series[1/(1-Sinh[x]-x-x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
A107404
Expansion of e.g.f. 1/(1 - sinh(x))^2.
Original entry on oeis.org
1, 2, 6, 26, 144, 962, 7536, 67706, 685824, 7730882, 95970816, 1300815386, 19113775104, 302616787202, 5135568746496, 92996021795066, 1789758460329984, 36479831022049922, 785020114093080576, 17785273588395966746, 423150055005134782464, 10548427254444904799042
Offset: 0
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E(x):=1/(1-sinh(x))^2: f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..30);
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CoefficientList[Series[1/(1-Sinh[x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
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a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (k+1)!*a136630(n, k)); \\ Seiichi Manyama, Feb 17 2025
A332256
E.g.f.: 1 / (2 - exp(sinh(x))).
Original entry on oeis.org
1, 1, 3, 14, 87, 672, 6231, 67412, 833475, 11593140, 179170947, 3045978388, 56490392943, 1134970258372, 24557211519951, 569294311105300, 14077429483372251, 369861235318338404, 10289111057247180411, 302132879478864660340, 9338874072977661538119
Offset: 0
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nmax = 20; CoefficientList[Series[1/(2 - Exp[Sinh[x]]), {x, 0, nmax}], x] Range[0, nmax]!
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seq(n)={Vec(serlaplace(1/(2 - exp(sinh(x + O(x*x^n))))))} \\ Andrew Howroyd, Feb 08 2020