A191277
Expansion of e.g.f. 1/(1 - sinh(x)*cosh(x)).
Original entry on oeis.org
1, 1, 2, 10, 56, 376, 3152, 30640, 338816, 4226176, 58564352, 892337920, 14834994176, 267186021376, 5182147684352, 107689460377600, 2387077442011136, 56219583797886976, 1401949974947889152, 36902741817196871680, 1022494706646806429696
Offset: 0
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CoefficientList[Series[1/(1-Sinh[x]*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
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a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k),k,1,n);
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my(x='x+O('x^30)); Vec(serlaplace(1/(1 - sinh(x)*cosh(x)))) \\ Michel Marcus, Jun 30 2022
A381277
Expansion of e.g.f. exp(sinh(3*x) / 3).
Original entry on oeis.org
1, 1, 1, 10, 37, 172, 1477, 8416, 74377, 683344, 5836969, 67102048, 699721453, 8268521536, 107106298093, 1347611617792, 19462095444241, 279380302430464, 4247519795325649, 68946703997616640, 1122787065355425973, 19697500164381137920, 351304020205694058133
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, 3^(n-k)*a136630(n, k));
A351891
Expansion of e.g.f. exp( sinh(sqrt(2)*x) / sqrt(2) ).
Original entry on oeis.org
1, 1, 1, 3, 9, 25, 105, 443, 1969, 10609, 57265, 338547, 2190969, 14498185, 104277849, 784965803, 6150938593, 51229928929, 440694547681, 3967606065891, 37247506348905, 361022009762809, 3645855348771273, 38001754007842715, 409302848055407761, 4558828622414199121
Offset: 0
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nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 2^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
A351892
Expansion of e.g.f. exp( sinh(sqrt(3)*x) / sqrt(3) ).
Original entry on oeis.org
1, 1, 1, 4, 13, 40, 205, 952, 4921, 31168, 189145, 1318528, 9843781, 74869888, 632536933, 5475991552, 49996774897, 485393809408, 4829958877105, 50858117779456, 554544498995965, 6259096187060224, 73822470722135293, 894846287081242624, 11261265009125680681, 146272258394568687616
Offset: 0
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nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[3] x]/Sqrt[3]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 3^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
A381343
Expansion of e.g.f. exp( sin(sqrt(2)*x) / sqrt(2) ).
Original entry on oeis.org
1, 1, 1, -1, -7, -15, 25, 287, 721, -2847, -30255, -61697, 682761, 5861713, 3105193, -258188513, -1681060063, 4623681473, 135471132705, 564325398271, -6357495670375, -89817656595791, -84337394884167, 7820620314702879, 67277670159083761, -322108989883888479
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, (-2)^((n-k)/2)*a136630(n, k));
Showing 1-5 of 5 results.