A032111
"BIJ" (reversible, indistinct, labeled) transform of 2,2,2,2...
Original entry on oeis.org
2, 6, 38, 366, 4502, 66606, 1149878, 22687566, 503589782, 12420052206, 336947795318, 9972186170766, 319727684645462, 11039636939221806, 408406422098722358, 16116066766061589966, 675700891505466507542
Offset: 1
A336217
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k)^2 * a(k).
Original entry on oeis.org
1, 2, 18, 362, 12946, 723402, 58208490, 6375093258, 911949196434, 165104835435146, 36903191037412618, 9980525774650881738, 3212329170232153022314, 1213419234370490738427722, 531582989226188067128503722, 267336170027296964096123899962
Offset: 0
-
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; CoefficientList[Series[1/(1 - 2 Sum[x^k/(k!)^2, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
A375991
Expansion of e.g.f. (3 - 2 * exp(x))^(3/2).
Original entry on oeis.org
1, -3, 0, 9, 45, 252, 1935, 19989, 260190, 4063887, 73823445, 1527002694, 35408499885, 909389617497, 25618701424680, 785355764569749, 26024092206299505, 926859918577582332, 35306305954587340515, 1432301360556686816529, 61649353087003554947550
Offset: 0
-
With[{nn=20},CoefficientList[Series[(3-2Exp[x])^(3/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)
-
a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*stirling(n, k, 2));
A384413
Expansion of Product_{k>=1} 1/(1 - k^3 * x)^((1/6) * (2/3)^k).
Original entry on oeis.org
1, 37, 33987, 169103895, 2499834885228, 81779253109721484, 5002571587280667349252, 513188808423273125116834036, 81795428604490137664191461936826, 19140816569244304756404266108586220066, 6295058477497449841660364475294196843864030, 2810342651288539045376339873565157506716615522598
Offset: 0
-
b(n) = sum(k=0, n, 2^k*k!*stirling(n, k, 2));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, b(3*k)*x^k/k)/2))