A318249
a(n) = (n - 1)! * d(n), where d(n) = number of divisors of n (A000005).
Original entry on oeis.org
1, 2, 4, 18, 48, 480, 1440, 20160, 120960, 1451520, 7257600, 239500800, 958003200, 24908083200, 348713164800, 6538371840000, 41845579776000, 2134124568576000, 12804747411456000, 729870602452992000, 9731608032706560000, 204363768686837760000, 2248001455555215360000, 206816133911079813120000
Offset: 1
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Table[(n - 1)! DivisorSigma[0, n], {n, 1, 24}]
nmax = 24; Rest[CoefficientList[Series[Sum[Sum[x^(j k)/(j k), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
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a(n) = (n-1)!*numdiv(n); \\ Michel Marcus, Aug 22 2018
A356564
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^x.
Original entry on oeis.org
1, 0, 2, 0, 28, -30, 888, -1260, 51728, -196560, 5293080, -22286880, 710229408, -4851269280, 138348035616, -1091188098000, 36482139114240, -379928382462720, 11812558481332992, -137793570801143040, 4609972759421554560, -67292912045817561600
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k))^x))
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a048272(n) = sumdiv(n, d, (-1)^(n/d+1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a048272(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
A338813
Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1+x^j)^(u/j).
Original entry on oeis.org
1, 0, 1, 4, 0, 1, -6, 16, 0, 1, 48, -30, 40, 0, 1, 0, 448, -90, 80, 0, 1, 1440, -840, 2128, -210, 140, 0, 1, -10080, 23532, -6720, 7168, -420, 224, 0, 1, 120960, -127008, 177868, -30240, 19488, -756, 336, 0, 1, 0, 2191104, -1018080, 892540, -100800, 45696, -1260, 480, 0, 1
Offset: 1
exp(Sum_{n>0} u*A048272(n)*x^n/n) = 1 + u*x + u^2*x^2/2! + (4*u+u^3)*x^3/3! + ... .
Triangle begins:
1;
0, 1;
4, 0, 1;
-6, 16, 0, 1;
48, -30, 40, 0, 1;
0, 448, -90, 80, 0, 1;
1440, -840, 2128, -210, 140, 0, 1;
-10080, 23532, -6720, 7168, -420, 224, 0, 1;
...
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a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
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{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1+x^j+x*O(x^n))^(u/j)), n), k)}
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a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))
Showing 1-3 of 3 results.
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