A333981
a(0) = 0; a(n) = 2^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 2^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 4, 34, 576, 16296, 691408, 41069568, 3252707328, 331218217600, 42159307194624, 6558777387076608, 1224428872399488000, 270143735036619436032, 69534931015726331203584, 20651854796028308275851264, 7009822878720340562163007488, 2696576146784893519040303235072, 1166999997199470676471689819258880
Offset: 0
-
a[0] = 0; a[n_] := a[n] = 2^(n - 1) + (1/n) Sum[Binomial[n, k]^2 2^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(3 - BesselI[0, 2 Sqrt[2 x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2
-
@CachedFunction
def a(n): return 0 if (n==0) else 2^(n-1) + (1/n)*sum(binomial(n,k)^2 *2^(k-1)*(n-k)*a(n-k) for k in (1..n-1)) # a= A333981
[a(n) for n in (0..30)] # G. C. Greubel, Jun 09 2022
A102222
Logarithm of triangular matrix A102220, which equals [2*I - A008459]^(-1).
Original entry on oeis.org
0, 1, 0, 3, 4, 0, 22, 27, 9, 0, 323, 352, 108, 16, 0, 7906, 8075, 2200, 300, 25, 0, 290262, 284616, 72675, 8800, 675, 36, 0, 14919430, 14222838, 3486546, 395675, 26950, 1323, 49, 0, 1022475715, 954843520, 227565408, 24793216, 1582700, 68992, 2352, 64, 0
Offset: 0
Rows begin:
[0],
[1,0],
[3,4,0],
[22,27,9,0],
[323,352,108,16,0],
[7906,8075,2200,300,25,0],
[290262,284616,72675,8800,675,36,0],...
which equals the term-by-term product of column 0
with the squared binomial coefficients (A008459):
[(0)1^2],
[(1)1^2,(0)1^2],
[(3)1^2,(1)2^2,(0)1^2],
[(22)1^2,(3)3^2,(1)3^2,(0)1^2],
[(323)1^2,(22)4^2,(3)6^2,(1)4^2,(0)1^2],...
A333982
a(0) = 0; a(n) = 3^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 3^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 5, 48, 909, 28836, 1371384, 91308708, 8106024861, 925225277004, 132007041682380, 23019553116101268, 4817014157800460664, 1191268407723761654964, 343706793228408937835772, 114423311913128119741898268, 43534429651349601213257298621, 18771927426013054800534345817884, 9106204442628918977341144456510260
Offset: 0
-
a[0] = 0; a[n_] := a[n] = 3^(n - 1) + (1/n) Sum[Binomial[n, k]^2 3^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(4 - BesselI[0, 2 Sqrt[3 x]])/3], {x, 0, nmax}], x] Range[0, nmax]!^2
A333983
a(0) = 0; a(n) = 4^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 4^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 6, 64, 1328, 46336, 2423040, 177379840, 17314109440, 2172895068160, 340868882825216, 65356107645583360, 15037174515952517120, 4088810357694136320000, 1297103066111891262668800, 474788193071044243776077824, 198617395218460028950533898240, 94165608216423156721014443868160
Offset: 0
-
a[0] = 0; a[n_] := a[n] = 4^(n - 1) + (1/n) Sum[Binomial[n, k]^2 4^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(5 - BesselI[0, 4 Sqrt[x]])/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
-
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
A333985
a(0) = 0; a(n) = 6^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 6^(k-1) * (n-k) * a(n-k).
Original entry on oeis.org
0, 1, 8, 102, 2448, 99576, 6070032, 517803840, 58901955840, 8614609282944, 1574889814326528, 351896788824053760, 94354291010501932032, 29899137879209196380160, 11053567519385396409446400, 4715135497874174650128617472, 2298676381054790419739595571200, 1270045124912998373344157769891840
Offset: 0
-
a[0] = 0; a[n_] := a[n] = 6^(n - 1) + (1/n) Sum[Binomial[n, k]^2 6^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(7 - BesselI[0, 2 Sqrt[6 x]])/6], {x, 0, nmax}], x] Range[0, nmax]!^2
A336437
a(n) = (n!)^n * [x^n] -log(1 - Sum_{k>=1} x^k / (k!)^n).
Original entry on oeis.org
0, 1, 3, 100, 104585, 5781843126, 25450069471437282, 12456703705462747095073458, 900677059707267544414220026068619393, 12337778954350678368447638232258657486399628887370, 39982077640755835496555968029419604779794754953051698069276656138
Offset: 0
-
Table[(n!)^n SeriesCoefficient[-Log[1 - Sum[x^k/(k!)^n, {k, 1, n}]], {x, 0, n}], {n, 0, 10}]
b[n_, k_] := If[n == 0, 0, 1 + (1/n) Sum[Binomial[n, j]^k j b[j, k], {j, 1, n - 1}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 10}]
A336427
a(0) = 0; a(n) = 1 + (1/n) * Sum_{k=1..n-1} binomial(n,k)^3 * k * a(k).
Original entry on oeis.org
0, 1, 5, 100, 5357, 597726, 120049592, 39381634818, 19686000625517, 14233714132535146, 14293760060523962630, 19298235276251711246358, 34108177389621376109912120, 77181320123960021972892515094, 219430688163572488543090308547898
Offset: 0
-
a[0] = 0; a[n_] := a[n] = 1 + (1/n) Sum[Binomial[n, k]^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[-Log[1 - Sum[x^k/(k!)^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^3
Showing 1-8 of 8 results.
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