A371399
a(n) = 2^n * Sum_{k=0..n} binomial(k + n, k) * binomial(2*n - k, n) * (-1/2)^k.
Original entry on oeis.org
1, 2, 12, 60, 340, 1932, 11256, 66264, 394020, 2359500, 14211912, 86004360, 522502344, 3184844600, 19467675120, 119288938800, 732508344516, 4506518476620, 27771180181800, 171393806476200, 1059200506065240, 6553715347503720, 40595235803924880, 251709010315822800
Offset: 0
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seq((2^n*add(binomial(k+n, k)*binomial(2*n-k, n)*(-1/2)^k, k=0..n)), n=0..23);
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a[n_] := 2^n Binomial[2 n, n] Hypergeometric2F1[-n, 1 + n, -2 n, -1/2];
Table[a[n], {n, 0, 23}]
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from math import comb
def A371399(n): return sum(comb(k+n,k)*comb((n<<1)-k,n)*(-1 if k&1 else 1)<Chai Wah Wu, Mar 22 2024
A371395
Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).
Original entry on oeis.org
1, 1, 1, 2, 3, 2, 5, 10, 10, 5, 14, 35, 45, 35, 14, 42, 126, 196, 196, 126, 42, 132, 462, 840, 1008, 840, 462, 132, 429, 1716, 3564, 4950, 4950, 3564, 1716, 429, 1430, 6435, 15015, 23595, 27225, 23595, 15015, 6435, 1430
Offset: 0
Triangle begins:
[0] [ 1],
[1] [ 1, 1],
[2] [ 2, 3, 2],
[3] [ 5, 10, 10, 5],
[4] [14, 35, 45, 35, 14],
[5] [42, 126, 196, 196, 126, 42].
Column 0 and main diagonal are
A000108.
Column 1 and subdiagonal are
A001700.
The even bisection of the alternating row sums is
A001764.
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T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # Peter Luschny, Mar 21 2024
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T[n_, k_] := (Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, k - n, 1, 1]) /(n + 1); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Peter Luschny, Mar 21 2024 *)
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def Trow(n):
return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]
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# As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
t = polygen(QQ, 't')
x = LazyPowerSeriesRing(t.parent(), 'x').0
gf = x*(1-x)*(1-t*x)
coeffs = gf.revert() / x
for n in range(6):
print(coeffs[n].list())
A371401
Triangle read by rows: T(n, k) = [x^k] (n*x + 1)*Hypergeometric([-n, -n + 1], [1], x).
Original entry on oeis.org
1, 1, 1, 1, 4, 4, 1, 9, 21, 9, 1, 16, 66, 76, 16, 1, 25, 160, 340, 205, 25, 1, 36, 330, 1100, 1275, 456, 36, 1, 49, 609, 2905, 5425, 3801, 889, 49, 1, 64, 1036, 6664, 18130, 20776, 9604, 1576, 64, 1, 81, 1656, 13776, 51156, 86436, 65856, 21456, 2601, 81
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 1, 4, 4;
[3] 1, 9, 21, 9;
[4] 1, 16, 66, 76, 16;
[5] 1, 25, 160, 340, 205, 25;
[6] 1, 36, 330, 1100, 1275, 456, 36;
[7] 1, 49, 609, 2905, 5425, 3801, 889, 49;
[8] 1, 64, 1036, 6664, 18130, 20776, 9604, 1576, 64;
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P := (n, x) -> (n*x + 1)*hypergeom([-n, -n + 1], [1], x):
T := (n, k) -> coeff(simplify(P(n, x)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
Showing 1-3 of 3 results.
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