Original entry on oeis.org
1, 23, 310, 3195, 27866, 216566, 1546028, 10338515, 65635570, 399429602, 2346750900, 13384232030, 74417751940, 404759481420, 2159510136408, 11327603405955, 58528412321250, 298354368109930, 1502525977613540
Offset: 0
-
R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) )); // G. C. Greubel, Jan 13 2020
-
seq(coeff(series((sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 13 2020
-
CoefficientList[Series[(Sqrt[1-4*x] +4*x-1)/(2*x*(1-4*x)^6), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
-
my(x='x+O('x^40)); Vec( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ) \\ G. C. Greubel, Jan 13 2020
-
def A045530_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ).list()
A045530_list(40) # G. C. Greubel, Jan 13 2020
A046527
A triangle related to A000108 (Catalan) and A000302 (powers of 4).
Original entry on oeis.org
1, 1, 1, 2, 5, 1, 5, 22, 9, 1, 14, 93, 58, 13, 1, 42, 386, 325, 110, 17, 1, 132, 1586, 1686, 765, 178, 21, 1, 429, 6476, 8330, 4746, 1477, 262, 25, 1, 1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1, 4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1
Offset: 0
Triangle begins as:
1;
1, 1;
2, 5, 1;
5, 22, 9, 1;
14, 93, 58, 13, 1;
42, 386, 325, 110, 17, 1;
132, 1586, 1686, 765, 178, 21, 1;
429, 6476, 8330, 4746, 1477, 262, 25, 1;
1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1;
4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1;
-
A046527:= func< n,k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
[A046527(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 28 2024
-
T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n,k-1]*(4^(n-k+ 1) -Binomial[2*n,n]/Binomial[2*(k-1),k-1])];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)
-
def A046527(n,k):
if k==0: return catalan_number(n)
else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
flatten([[A046527(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 28 2024
Original entry on oeis.org
1, 25, 362, 3973, 36646, 299530, 2238676, 15613741, 103054094, 650194974, 3950996556, 23257207714, 133217073276, 745218012084, 4083224828328, 21966983072637, 116268166691358, 606474982072982, 3122157367765788
Offset: 1
-
R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(1-4*x))/(2*(1-4*x)^6) )); // G. C. Greubel, Jan 13 2020
-
seq(coeff(series((1-sqrt(1-4*x))/(2*(1-4*x)^6), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 13 2020
-
CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-4*x)^6), {n,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
-
my(x='x+O('x^40)); Vec((1-sqrt(1-4*x))/(2*(1-4*x)^6)) \\ G. C. Greubel, Jan 13 2020
-
def A045622_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1-sqrt(1-4*x))/(2*(1-4*x)^6) ).list()
A045622_list(40) # G. C. Greubel, Jan 13 2020
Showing 1-3 of 3 results.
Comments