A362585
Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).
Original entry on oeis.org
1, 1, 1, 3, 6, 3, 13, 39, 39, 13, 75, 300, 450, 300, 75, 541, 2705, 5410, 5410, 2705, 541, 4683, 28098, 70245, 93660, 70245, 28098, 4683, 47293, 331051, 993153, 1655255, 1655255, 993153, 331051, 47293, 545835, 4366680, 15283380, 30566760, 38208450, 30566760, 15283380, 4366680, 545835
Offset: 0
[0] 1;
[1] 1, 1;
[2] 3, 6, 3;
[3] 13, 39, 39, 13;
[4] 75, 300, 450, 300, 75;
[5] 541, 2705, 5410, 5410, 2705, 541;
[6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;
-
def TransOrdPart(m, n) -> list[int]:
@cached_function
def P(m: int, n: int):
R = PolynomialRing(ZZ, "x")
if n == 0: return R(1)
return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
for k in range(1, n + 1)))
T = P(m, n)
def C(k) -> int:
return sum(T[j] * binomial(n, k) for j in range(n + 1))
return [C(k) for k in range(n+1)]
def A362585(n) -> list[int]: return TransOrdPart(1, n)
for n in range(6): print(A362585(n))
A346432
a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.
Original entry on oeis.org
1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000
Offset: 0
Cf.
A000670,
A001339,
A002866,
A003480,
A007840,
A052555,
A052567,
A136658,
A216794,
A308939,
A346433.
-
a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
-
my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021
A346433
E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).
Original entry on oeis.org
1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470
Offset: 0
-
nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
-
my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021
A382753
Expansion of e.g.f. 3/(5 - 2*exp(3*x)).
Original entry on oeis.org
1, 2, 14, 138, 1806, 29562, 580734, 13309578, 348611886, 10272416922, 336326121054, 12112707922218, 475894244100366, 20255443904321082, 928448378212678974, 45597074777924954058, 2388608236671667179246, 132947999835258872046042, 7835059049893316949502494
Offset: 0
A337027
a(n) = (1/2) * Sum_{k>=0} (2*k + n)^n / 2^k.
Original entry on oeis.org
1, 3, 24, 293, 4784, 97687, 2393472, 68405073, 2233928448, 82063263371, 3349249267712, 150353137462717, 7362889615257600, 390601858379350815, 22315011551291080704, 1365896953310909493929, 89179296762081886011392, 6186383336743041502051219
Offset: 0
-
Table[2^(n - 1) HurwitzLerchPhi[1/2, -n, n/2], {n, 0, 17}]
Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[2 x]), {x, 0, n}], {n, 0, 17}]