A295113
a(n) = (1/n)*Sum_{k=0..n-1} (8*k + 9)*A295112(k)^2.
Original entry on oeis.org
9, 13, 17, 219, 1561, 8169, 39321, 191389, 985201, 5430789, 31943961, 198471183, 1288665177, 8665236121, 59922226809, 423935337411, 3056528058577, 22392246851973, 166311049602681, 1250027314777795, 9494339129623329, 72784922204637153, 562626619763553217, 4381665416129531961, 34354964747652467697
Offset: 1
a(2) = 13 since (1/2)*Sum_{k=0..1} (8k + 9)*A295112(k)^2 = (1/2)*((8*0 + 9)*A295112(0)^2 + (8 + 9)*A295112(1)^2) = (1/2)*(9*(-1)^2 + 17*(-1)^2) = 13.
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W[n_]:=W[n]=Sum[Binomial[n,2k]Binomial[2k,k]/(2k-1),{k,0,n/2}];
a[n_]:=a[n]=1/n*Sum[(8k+9)W[k]^2,{k,0,n-1}];
Table[a[n],{n,1,25}]
A295132
a(n) = (2/n)*Sum_{k=1..n} (2k+1)*M(k)^2 where M(k) is the Motzkin number A001006(k).
Original entry on oeis.org
6, 23, 90, 432, 2286, 13176, 80418, 513764, 3400518, 23167311, 161640554, 1150633512, 8332048638, 61232315553, 455830692210, 3432015694314, 26101221114582, 200295455169015, 1549473966622602, 12074304397434552, 94713783502786686, 747454269790900728
Offset: 1
a(2) = 23 since (2/2)*Sum_{k=1..2} (2k + 1)*M(k)^2 = (2*1 + 1)*M(1)^2 + (2*2 + 1)*M(2)^2 = 3*1^2 + 5*2^2 = 23.
- Zhi-Wei Sun, Table of n, a(n) for n = 1..200
- Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
- Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schroder numbers, J. Number Theory 183(2018), 146-171.
- Zhi-Wei Sun, On Motzkin numbers and central trinomial coefficients, arXiv:1801.08905 [math.CO], 2018.
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h := k -> (4*k+2)*hypergeom([(1-k)/2,-k/2],[2],4)^2:
a := proc(n) add(simplify(h(k)),k=1..n): if % mod n = 0 then %/n else -1 fi end:
seq(a(n), n=1..25); # Peter Luschny, Nov 16 2017
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M[n_] := M[n] = Sum[Binomial[n, 2k] Binomial[2k, k]/(k + 1), {k, 0, n/2}];
a[n_] := a[n] = 2/n * Sum[(2k + 1) M[k]^2, {k, 1, n}];
Table[a[n], {n, 1, 25}]
Original entry on oeis.org
1, 3, 8, 20, 51, 133, 356, 972, 2695, 7557, 21372, 60840, 174097, 500295, 1442720, 4172752, 12099411, 35161001, 102375400, 298586652, 872177273, 2551118623, 7471195500, 21904500500, 64286141881, 188844619563, 555216323396, 1633658183432, 4810340397375, 14173698242137
Offset: 0
Cf.
A002144,
A002145,
A002426,
A024997,
A025178,
A026083,
A097893,
A128014,
A247287,
A295112,
A383527.
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a := series(exp(x)*(BesselI(0, 2*x) + 2*int(BesselI(0, 2*x), x) + int(int(BesselI(0, 2*x), x), x)), x = 0, 30): seq(n!*coeff(a, x, n), n = 0 .. 29);
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a(n) = sum(k=0, n, sum(i=0, k, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))); \\ Michel Marcus, Aug 06 2025
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from math import comb as C
def a(n):
return sum(C(n+1, k+1)*C(2*(k//2), k//2) for k in range(n + 1))
print([a(n) for n in range(30)])
Showing 1-3 of 3 results.
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