A295553
Expansion of 1/(1 - Sum_{k>=1} (2*k-1)!!*x^k).
Original entry on oeis.org
1, 1, 4, 22, 154, 1330, 13882, 171802, 2474098, 40738594, 755322778, 15566915770, 352862768434, 8720662458754, 233285616212506, 6713983428179098, 206813607458357746, 6788092999359053410, 236481982146071359258, 8714521818620631672058, 338660320676350494328882, 13841377309645038610883266
Offset: 0
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
- Index entries for sequences related to factorial numbers
- Index entries for sequences related to compositions
-
nmax = 21; CoefficientList[Series[1/(1 - Sum[(2 k - 1)!! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1 + x/(1 - 2 x + ContinuedFractionK[-k x, 1, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
A307364
Expansion of 1/(1 - Sum_{k>=1} prime(k)#*x^k), where prime(k)# is the product of first k primes (A002110).
Original entry on oeis.org
1, 2, 10, 62, 454, 4310, 49954, 746078, 13180750, 283749638, 7747573666, 234558524690, 8437098259486, 340293472077722, 14523592739559970, 676119676949381762, 35425760935764788014, 2070535245695282709950, 125884029549845876309674, 8379955313909510350628018
Offset: 0
-
nmax = 19; CoefficientList[Series[1/(1 - Sum[Product[Prime[j], {j, k}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Product[Prime[j], {j, k}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
A370378
Number of compositions of n where there are (2*k)! sorts of part k.
Original entry on oeis.org
1, 2, 28, 824, 44080, 3837536, 496714432, 89388391808, 21308786907904, 6492490191541760, 2459980247094946816, 1134165248844198336512, 625104522913814858149888, 405845822590303335956701184, 306541019968859037778756157440
Offset: 0
-
nmax = 15; CoefficientList[Series[1/(1 - Sum[(2*k)! * x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 17 2024 *)
-
my(N=20, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, (2*k)!*x^k)))
A305535
Expansion of 1/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 4*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.
Original entry on oeis.org
1, 1, 3, 13, 75, 557, 5179, 58589, 784715, 12154061, 213593563, 4195613373, 91031201643, 2160916171181, 55687501548539, 1547866851663261, 46150908197995403, 1469089501918434957, 49722765216242122267, 1782934051704982201469, 67514992620138056010667
Offset: 0
-
nmax = 20; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-2 Floor[(k + 1)/2] x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 20; CoefficientList[Series[1/(1 - Sum[2^(k - 1) (k - 1)! x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^(k - 1) (k - 1)! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Comments