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A365028
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a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n,k)*binomial(n+k-1,n)* binomial(3*n+k-1,n).
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1
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1, 3, 33, 462, 7185, 118503, 2029650, 35690868, 639948177, 11647493715, 214523842533, 3989404547076, 74784662259762, 1411371612900018, 26789659159105260, 511034151538808712, 9790719515677254033, 188293669308690649515, 3633506906803796715585
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OFFSET
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0,2
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COMMENTS
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Compare with the binomial sum identities:
1) Sum_{k = 0..n} (-1)^(n+k) * binomial(n,k)*binomial(n+k-1,n)* binomial(2*n+k-1,n) = (1/2)*binomial(2*n,n)^2 = (1/2)*A002894(n) for n >= 1 and
2) Sum_{k = 0..2*n} (-1)^k * binomial(2*n,k)*binomial(n+k-1,n)*binomial(2*n+k-1,n) = binomial(2*n,n) = A000984(n).
More generally, for x != -1, it appears that the identity Sum_{k = 0..n} (-1)^(n+k) * binomial(n,k)*binomial(x*n+k-1,n)*binomial((x+1)*n+k-1,n) = x/(x+1) * binomial((x+1)*n,n)*binomial(2*n,n) holds, and also, for arbitrary x and y, the identity Sum_{k = 0..2*n} (-1)^k * binomial(2*n,k)*binomial(x*n+k-1,n)* binomial(y*n+k-1,n) = binomial(2*n,n) appears to hold.
The central binomial coefficients A000984(n) satisfy the supercongruences A000984(n*p^r) == A000984(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r (see Meštrović, equation 39). We conjecture that the present sequence satisfies the same supercongruences.
More generally, for M a positive integer and integers A, B and C, we conjecture that the sequence whose n-th term is given by Sum_{k = 0..M*n} (-1)^(n+k) * binomial(A*n,k)*binomial(B*n+k-1,n)*binomial(C*n+k-1,n) also satisfies the same supercongruences.
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LINKS
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FORMULA
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P-recursive:
n^2*(3*n - 1)*(3*n - 2)*(3*n - 3)*(143*n^4 - 891*n^3 + 2080*n^2 - 2154*n + 834)*a(n) = (n - 1)*(83941*n^8 - 690899*n^7 + 2387290*n^6 - 4496969*n^5 + 5024203*n^4 - 3396638*n^3 + 1355652*n^2 - 292104*n + 25920)*a(n-1) - 9*(n - 2)*(3*n - 4)^2*(3*n - 5)^2*(143*n^4 - 319*n^3 + 265*n^2 - 95*n + 12)*a(n-2) with a(0) = 1 and a(1) = 3.
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MAPLE
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seq( add( (-1)^(n+k) * binomial(n, k)*binomial(n+k-1, n)*binomial(3*n+k-1, n), k = 0..n) ), n = 0..20);
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MATHEMATICA
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A365028[n_]:=Sum[(-1)^(n+k)Binomial[n, k]Binomial[n+k-1, n]Binomial[3n+k-1, n], {k, 0, n}]; Array[A365028, 30, 0] (* Paolo Xausa, Oct 06 2023 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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