A306347
Expansion of e.g.f. exp((sin(x) + sinh(x))/2).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 389, 1904, 9329, 38040, 132147, 542648, 3283633, 20997824, 114657097, 536178880, 2784500161, 19876061312, 153326461311, 1034551839872, 6051063485481, 38079448046208, 312420426154893, 2785055242928768, 22141255520251313
Offset: 0
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nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ Seiichi Manyama, Mar 17 2022
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a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ Seiichi Manyama, Mar 17 2022
A365911
Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+3) / (4*k+3)! ).
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 20, 1, 0, 1680, 240, 1, 369600, 102960, 4160, 168168001, 76876800, 7743840, 137225153280, 93117024001, 17091609600, 182510023324320, 172080261401600, 49615854288001, 369403226582016000, 461748751736204400, 191552892427653120
Offset: 0
A013369
Expansion of e.g.f. exp(sin(x)-sinh(x)).
Original entry on oeis.org
1, 0, 0, -2, 0, 0, 40, -2, 0, -2240, 480, -2, 246400, -137280, 8320, -44844802, 51251200, -10325120, 12197916160, -24831206402, 11394406400, -4636033573760, 15296025241600, -13230894476802, 2348235343872000
Offset: 0
Patrick Demichel (patrick.demichel(AT)hp.com)
1-2/3!*x^3+40/6!*x^6-2/7!*x^7-2240/9!*x^9...
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With[{nn=30},CoefficientList[Series[Exp[Sin[x]-Sinh[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 13 2020 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sin(x)-sinh(x)))) \\ Seiichi Manyama, Mar 17 2022
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a(n) = if(n==0, 1, -2*sum(k=0, (n-3)\4, binomial(n-1, 4*k+2)*a(n-4*k-3))); \\ Seiichi Manyama, Mar 17 2022
A307979
Expansion of e.g.f. exp((cosh(x) - cos(x))/2) (even powers only).
Original entry on oeis.org
1, 1, 3, 16, 133, 1576, 24783, 495496, 12245353, 364768576, 12838252443, 526095538816, 24781014246253, 1326767681420416, 80013978835916583, 5392682199766283776, 403287063337529642833, 33261775377836063850496, 3009257393136250807614003, 297176659119237977183973376
Offset: 0
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nmax = 19; Table[(CoefficientList[Series[Exp[(Cosh[x] - Cos[x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[n_] := a[n] = Sum[Boole[MemberQ[{2}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[2 n], {n, 0, 19}]
A365893
Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+3) / (5*k+3)! ).
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 10, 0, 1, 280, 0, 165, 15400, 1, 30030, 1401400, 6995, 6806800, 190590401, 6506835, 1939938000, 36212380820, 4940624150, 687126039601, 9163671323015, 3761116975000, 297754623925175, 2982764271647875, 3067236941769001
Offset: 0
Showing 1-5 of 5 results.
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