A337824
a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).
Original entry on oeis.org
0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600
Offset: 0
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S:= series(log(1+x*BesselI(0,2*sqrt(x))),x,31):
0,seq(coeff(S,x,n)*(n!)^2, n=1..30); # Robert Israel, Jan 07 2024
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a[0] = 0; a[n_] := a[n] = n^2 - (1/n) * Sum[(Binomial[n, k] (n - k))^2 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Log[1 + x BesselI[0, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
A337826
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).
Original entry on oeis.org
1, 1, 10, 105, 2248, 62445, 2390436, 116650177, 7043659904, 514744959321, 44534754680500, 4493090921151261, 521600149636044480, 68900819660071184149, 10259571068808850618480, 1708054303772376318547125, 315688007001129064574027776, 64370788231256983836207599153
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k^4 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2
A342182
Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - x * BesselI(0,2*sqrt(x))).
Original entry on oeis.org
1, 1, 8, 117, 3184, 134025, 8141436, 672837277, 72634878016, 9923765772177, 1673881314096700, 341631408064928421, 82978986493779894288, 23653894531273155603961, 7819996460332550715977588, 2967815528758036870644773925, 1281517958938232539844046259456
Offset: 0
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nmax = 16; CoefficientList[Series[1/(1 - x BesselI[0, 2 Sqrt[x]]), {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(Binomial[n, k] (n - k))^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
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a(n) = {n!^2*polcoef(1 / (1 - sum(k=1, n, x^k / ((k-1)!)^2) + O(x*x^n)), n)} \\ Andrew Howroyd, Mar 04 2021
A352658
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).
Original entry on oeis.org
1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[k + 1, 2] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(x BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
Showing 1-4 of 4 results.