A344098
a(n) = [x^n] Product_{k>=1} (1 + x^k)^binomial(k+n-1,n-1).
Original entry on oeis.org
1, 1, 4, 29, 221, 2027, 21022, 242209, 3060262, 41936745, 618154670, 9735013136, 162892047930, 2882449728121, 53727527279464, 1051276401060921, 21529017626095851, 460231878244308738, 10246160509840187387, 237067632496414877363, 5689786581042000827057, 141415234722601777758232
Offset: 0
-
Table[SeriesCoefficient[Product[(1 + x^k)^Binomial[k + n - 1, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 21}]
A[n_, k_] := A[n, k] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(j/d + 1) d Binomial[d + k - 1, k - 1], {d, Divisors[j]}] A[n - j, k], {j, 1, n}]]; a[n_] := A[n, n]; Table[a[n], {n, 0, 21}]
A305205
a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).
Original entry on oeis.org
1, -1, -2, -3, -4, 30, 274, 1841, 9358, 32463, -41557, -2265846, -28939286, -272101778, -2038274408, -10494221259, 9056975574, 1244820826687, 22703501504125, 299864024917632, 3221417281127823, 26849622543478562, 110101743392268978, -1810492304600468063
Offset: 0
-
Table[SeriesCoefficient[Exp[-Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]
A344099
Expansion of Product_{k>=1} (1 + x^k)^binomial(k+3,4).
Original entry on oeis.org
1, 1, 5, 20, 60, 190, 561, 1651, 4720, 13300, 36716, 99872, 267836, 708890, 1854255, 4796273, 12279445, 31135188, 78236006, 194921680, 481758832, 1181675902, 2877646681, 6959866116, 16723591530, 39934902812, 94795718409, 223741936855, 525206126933, 1226393510220
Offset: 0
-
nmax = 29; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 3, 4], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 3, 4], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 29}]
A344100
Expansion of Product_{k>=1} (1 + x^k)^binomial(k+4,5).
Original entry on oeis.org
1, 1, 6, 27, 92, 323, 1070, 3527, 11314, 35708, 110478, 336629, 1011097, 2997233, 8778761, 25424358, 72867447, 206804742, 581573340, 1621407554, 4483701126, 12303384015, 33514076529, 90656680725, 243603875523, 650444927010, 1726229294595, 4554686670838, 11950683658941
Offset: 0
-
nmax = 28; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 4, 5], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 4, 5], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]
A344101
Expansion of Product_{k>=1} (1 + x^k)^binomial(k+5,6).
Original entry on oeis.org
1, 1, 7, 35, 133, 511, 1869, 6797, 24095, 83938, 286734, 964348, 3196984, 10460310, 33813984, 108076908, 341821250, 1070484009, 3321584021, 10217036263, 31169524988, 94351439060, 283498600776, 845848778722, 2506779443603, 7381617323598, 21603241378334, 62853440151768
Offset: 0
-
nmax = 27; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 5, 6], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 5, 6], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 27}]
A305654
a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^n)).
Original entry on oeis.org
1, 1, 4, 14, 65, 323, 1890, 12002, 83901, 630818, 5081318, 43546333, 395422430, 3788368227, 38151667046, 402516707510, 4436230390977, 50948789415297, 608433141666219, 7540823673023319, 96826154085714992, 1285991546051286085, 17640769457638701839, 249602608552024560609
Offset: 0
-
Table[SeriesCoefficient[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]
A305655
a(n) = [x^n] exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^n)).
Original entry on oeis.org
1, 1, 3, 13, 54, 290, 1674, 10857, 76398, 580230, 4706734, 40598349, 370694845, 3569027696, 36100349833, 382360758863, 4228730647420, 48716663849192, 583403253712747, 7248883337962522, 93291181556742684, 1241632098163126324, 17064777292709034968, 241874821482784132204
Offset: 0
-
Table[SeriesCoefficient[Exp[Sum[(-1)^(k + 1) x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 + x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]
A305255
a(n) = [x^n] exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 - x^k)^n)).
Original entry on oeis.org
1, -1, -1, -4, -3, 14, 240, 1686, 9479, 36761, 3412, -1951731, -27296124, -268495319, -2093667873, -11586874946, -3788945531, 1127535019748, 21900095232973, 297591401221473, 3270627818325128, 28116733997044842, 129815302615081267, -1568168714539146596, -59839621829784309343
Offset: 0
-
Table[SeriesCoefficient[Exp[Sum[(-1)^k x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[1/(1 + x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Showing 1-8 of 8 results.