A337597
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 6^(k-1) * a(n-k).
Original entry on oeis.org
1, 1, 8, 96, 1896, 55416, 2182752, 111162528, 7088997888, 550749341952, 51058009732608, 5556160183592448, 699989463219105792, 100917906076208203776, 16486415052067886690304, 3026039346413717945757696, 619431153899977856767131648, 140491838894751995366936641536, 35102748598142373142198776889344
Offset: 0
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S:= series(exp((BesselI(0,2*sqrt(6*x))-1)/6),x,51):
seq(coeff(S,x,j)*(j!)^2, j=0..50); # Robert Israel, Sep 06 2020
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 6^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[6 x]] - 1)/6], {x, 0, nmax}], x] Range[0, nmax]!^2
A337592
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 2^(k-1) * a(n-k).
Original entry on oeis.org
1, 1, 4, 28, 312, 4936, 104128, 2806336, 93560064, 3765265408, 179415074304, 9964625629696, 636737424291840, 46303081167540224, 3796275000959266816, 348100339275620651008, 35448445862069986361344, 3984266642444252234153984, 491556877841462376382332928
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[2 x]] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!^2
A337593
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 3^(k-1) * a(n-k).
Original entry on oeis.org
1, 1, 5, 42, 573, 11226, 294804, 9946791, 417064365, 21187915362, 1278636342660, 90195692894451, 7338668846348844, 681052861293535251, 71405270562056271741, 8388541745045127600597, 1096298129481068449931085, 158383969954582566159384786, 25153555538082783169267336764
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[3 x]] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!^2
A337594
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 4^(k-1) * a(n-k).
Original entry on oeis.org
1, 1, 6, 58, 920, 21176, 654960, 26114768, 1298070912, 78359732608, 5630565514496, 473796572027648, 46060380961356800, 5114737212582603776, 642502387594286036992, 90542358999393528670208, 14209873001490130067095552, 2467784343879850163370295296, 471558856613839054976849608704
Offset: 0
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a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 4 Sqrt[x]] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!^2
A333984
a(0) = 0; a(n) = 5^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 5^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 7, 82, 1839, 69630, 3950650, 313747050, 33224570175, 4523562983350, 769859662962750, 160137417877796250, 39971947204607486250, 11791483690935887486250, 4058152793413483423916250, 1611522009185095020022068750, 731368135285580087866788609375, 376178084508304435598172207843750
Offset: 0
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a[0] = 0; a[n_] := a[n] = 5^(n - 1) + (1/n) Sum[Binomial[n, k]^2 5^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(6 - BesselI[0, 2 Sqrt[5 x]])/5], {x, 0, nmax}], x] Range[0, nmax]!^2
Showing 1-5 of 5 results.