Search: a337591 -id:a337591
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A337824
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a(0) = 0; a(n) = n^2 - (1/n) * Sum_{k=1..n-1} (binomial(n,k) * (n-k))^2 * k * a(k).
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+10
2
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0, 1, 2, -15, 16, 2505, -60264, -606515, 131316928, -4813100271, -339213768200, 62401665573621, -2075963863814928, -745086903175541927, 140250562903680456332, 808225064553580739325, -5491409141464496462591744, 1013058261721909845376508449, 127689148764914765889971316600
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OFFSET
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0,3
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LINKS
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FORMULA
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Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * BesselI(0,2*sqrt(x))).
Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^2 * x^n / (n!)^2).
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MAPLE
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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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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A337826
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a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k^4 * a(n-k).
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+10
1
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1, 1, 10, 105, 2248, 62445, 2390436, 116650177, 7043659904, 514744959321, 44534754680500, 4493090921151261, 521600149636044480, 68900819660071184149, 10259571068808850618480, 1708054303772376318547125, 315688007001129064574027776, 64370788231256983836207599153
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OFFSET
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0,3
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LINKS
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FORMULA
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Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} n^3 * x^n / (n!)^2).
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A342182
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Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - x * BesselI(0,2*sqrt(x))).
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+10
0
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1, 1, 8, 117, 3184, 134025, 8141436, 672837277, 72634878016, 9923765772177, 1673881314096700, 341631408064928421, 82978986493779894288, 23653894531273155603961, 7819996460332550715977588, 2967815528758036870644773925, 1281517958938232539844046259456
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OFFSET
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0,3
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LINKS
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FORMULA
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Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / (1 - Sum_{n>=1} x^n / ((n-1)!)^2).
a(0) = 1; a(n) = Sum_{k=0..n-1} (binomial(n,k) * (n-k))^2 * a(k).
a(n) ~ n!^2 / ((1 + r^(3/2)*BesselI(1, 2*sqrt(r))) * r^n), where r = 0.592860029867912878114616561736048937618032595935338954527835... is the root of the equation r*BesselI(0, 2*sqrt(r)) = 1. - Vaclav Kotesovec, May 04 2024
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MATHEMATICA
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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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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A352658
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a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).
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+10
0
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1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359
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OFFSET
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0,3
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LINKS
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FORMULA
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Sum_{n>=0} a(n) * x^n / n!^2 = exp( (x * BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x))) / 2 ).
Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} binomial(n+1,2) * x^n / n!^2 ).
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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