Displaying 1-10 of 11 results found.
Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).
+10
7
1, 1, 5, 42, 498, 7644, 144156, 3225648, 83536008, 2457701928, 80970232104, 2953056534768, 118112744060208, 5140622709134496, 241863782829704928, 12232551538417012992, 661818290353375962240, 38140594162828447248000, 2332567001993176540206720, 150880256846462633823648000
FORMULA
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007559(k). a(n) ~ n! * exp(n/3) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021 a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
MAPLE
g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
MATHEMATICA
nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]
Expansion of e.g.f. 1 / (1 + 3 * log(1-x))^(2/3).
+10
6
1, 2, 12, 114, 1482, 24468, 490020, 11538840, 312363720, 9556741440, 326076452640, 12275391192480, 505400508041760, 22590511357965120, 1089423938332883520, 56379459359942190720, 3116574045158647605120, 183271869976364873222400
FORMULA
a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j+2)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} (3 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/3) * n^(n + 1/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1/3) - 1)^(n + 2/3) * exp(2*n/3)). - Vaclav Kotesovec, Nov 11 2023
MATHEMATICA
a[n_] := Sum[Product[3*j + 2, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
PROG
(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 3*j+2)*abs(stirling(n, k, 1)));
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).
+10
5
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0
FORMULA
E.g.f. of column k: 1/(1 + k*log(1 - x)).
A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022
EXAMPLE
E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 10, 21, 36, 55, ...
0, 14, 76, 222, 488, 910, ...
0, 88, 772, 3132, 8824, 20080, ...
0, 694, 9808, 55242, 199456, 553870, ...
MATHEMATICA
Table[Function[k, n! SeriesCoefficient[1/(1 + k Log[1 - x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(4/3).
+10
4
1, 4, 32, 372, 5652, 105936, 2360712, 60956472, 1789413864, 58850914752, 2143354213728, 85629122177760, 3723269780412000, 175035687610956480, 8846458578801144000, 478330017277120767360, 27551501517174431852160, 1684176901225092936990720
FORMULA
a(n) = Sum_{k=0..n} A007559(k+1) * |Stirling1(n,k)|.
MATHEMATICA
nmax=17; CoefficientList[Series[1 / (1 + 3 * Log[1-x])^(4/3), {x, 0, nmax}], x]*Range[0, nmax]! (* Stefano Spezia, Sep 03 2024 *)
PROG
(PARI) a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k+1)*abs(stirling(n, k, 1)));
Expansion of e.g.f. 1/(1 - x)^(3/(1 + 3 * log(1-x))).
+10
3
1, 3, 30, 438, 8334, 194580, 5368662, 170591022, 6126386724, 245127214548, 10804866210648, 519910458588576, 27105081897342816, 1521393008601586536, 91445577404393807928, 5858664681621903625368, 398467273528657973600208, 28668189882264862351707504
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} A354263(k) * binomial(n-1,k-1) * a(n-k). a(n) = Sum_{k=0..n} 3^k * A000262(k) * |Stirling1(n,k)|. a(n) ~ exp((-5 + 1/(exp(1/3) - 1) + 4*sqrt(3*n/(exp(1/3) - 1)) - 4*n)/6) * n^(n - 1/4) / (sqrt(2) * 3^(1/4) * (exp(1/3) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(3/(1+3*log(1-x)))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v;
Expansion of e.g.f. 1 / (1 - log(1 + 3*x) / 3).
+10
3
1, 1, -1, 6, -48, 534, -7542, 129240, -2603736, 60292512, -1577546928, 46021512096, -1480976147664, 52110720451152, -1990258155061776, 81995762243700864, -3624527727510038784, 171109526616468957312, -8591991935936929932672, 457246520477143117555968
FORMULA
a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 3^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-3)^(k-1) * a(n-k).
MATHEMATICA
nmax = 19; CoefficientList[Series[1/(1 - Log[1 + 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 3^(n - k), {k, 0, n}], {n, 0, 19}]
PROG
(PARI) my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+3*x)/3))) \\ Michel Marcus, Jun 06 2022
Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(5/3).
+10
3
1, 5, 45, 570, 9270, 183840, 4299360, 115795920, 3528915840, 120032889840, 4507313333040, 185185602462240, 8262852630732000, 397873645339668480, 20563762111640910720, 1135441077379757372160, 66703342626913255770240, 4154100873615633462894720
FORMULA
a(n) = (1/2) * Sum_{k=0..n} A008544(k+1) * |Stirling1(n,k)|.
MATHEMATICA
nmax=17; CoefficientList[Series[1 / (1 + 3 * Log[1-x])^(5/3), {x, 0, nmax}], x]*Range[0, nmax]! (* Stefano Spezia, Sep 03 2024 *)
PROG
(PARI) a008544(n) = prod(k=0, n-1, 3*k+2);
a(n) = sum(k=0, n, a008544(k+1)*abs(stirling(n, k, 1)))/2;
Expansion of e.g.f. (1 + 3 * log(1 - x))^(4/3).
+10
2
1, -4, 0, 12, 108, 1104, 14136, 225768, 4386168, 100885248, 2683789344, 81047258208, 2737919298528, 102266990392896, 4184016413001408, 186047367206499072, 8933002185371731200, 460580247564830138880, 25378595790818821816320, 1488230641037882346324480
FORMULA
a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-4)) * |Stirling1(n,k)|.
PROG
(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-4)*abs(stirling(n, k, 1)));
Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^2.
+10
1
1, 6, 60, 822, 14238, 297684, 7286076, 204251328, 6450932448, 226613038608, 8763294140064, 369900822475728, 16922169163019088, 833991953707934496, 44050579327333028448, 2482381132145285334912, 148660444826262311114880, 9427874254540824544312320
FORMULA
a(n) = Sum_{k=0..n} 3^k * (k+1)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * exp(2*n/3) * (exp(1/3) - 1)^(n+2)). - Vaclav Kotesovec, Sep 06 2024
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^2))
(PARI) a(n) = sum(k=0, n, 3^k*(k+1)!*abs(stirling(n, k, 1)));
Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^3.
+10
1
1, 9, 117, 1962, 40122, 966276, 26755812, 836862192, 29167596504, 1120629465432, 47044646845848, 2142210019297680, 105154320625284240, 5534780654854980000, 310945503593770489440, 18570787974013838515200, 1174884522886771261079040
FORMULA
a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
PROG
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^3))
(PARI) a(n) = sum(k=0, n, 3^k*(k+2)!*abs(stirling(n, k, 1)))/2;
Search completed in 0.005 seconds
|