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 A361884 a(n) = (1/n) * Sum_{k = 0..n} (-1)^(n+k) * (n + 2*k) * binomial(n+k-1,k)^3. 3
 2, 66, 2540, 110530, 5197752, 257490156, 13238524728, 699822144450, 37800431926400, 2077184897317816, 115757876008359312, 6526739641107783916, 371641758587326581200, 21341134886976332825400, 1234474507620634579565040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Compare with the closed form evaluation of the binomial sums (1/n) * Sum_{k = 0..n} (-1)^(n+k) * (n + 2*k) * binomial(n+k-1,k) = binomial(2*n,n) and (1/n) * Sum_{k = 0..n} (n + 2*k) * binomial(n+k-1,k)^2 = binomial(2*n,n)^2. The central binomial coefficients u(n) := binomial(2*n,n) = A000984(n) satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for positive integers n and r and all primes p >= 5. We conjecture that the present sequence satisfies the same congruences. More generally, for m >= 3, the sequences {b_m(n) : n >= 1} and {c_m(n) : n >= 1} defined by b_m(n) = (1/n) * Sum_{k = 0..n} (n + 2*k) * binomial(n+k-1,k)^m and c_m(n) = (1/n) * Sum_{k = 0..n} (-1)^k * (n + 2*k) * binomial(n+k-1,k)^m may both satisfy the same congruences. LINKS Table of n, a(n) for n=1..15. FORMULA a(n) ~ 2^(6*n) / (3 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 29 2023 MAPLE seq( (1/n)*add((-1)^(n+k) * (n + 2*k) * binomial(n+k-1, n-1)^3, k = 0..n), n = 1..20); MATHEMATICA Table[Sum[(-1)^(n+k) * (n + 2*k) * Binomial[n+k-1, k]^3, {k, 0, n}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Mar 29 2023 *) PROG (PARI) a(n) = (1/n) * sum(k = 0, n, (-1)^(n+k) * (n + 2*k) * binomial(n+k-1, k)^3); \\ Michel Marcus, Mar 30 2023 CROSSREFS Cf. A000984, A002894, A361883, A361885, A361886. Sequence in context: A218433 A092884 A230735 * A261784 A350730 A255435 Adjacent sequences: A361881 A361882 A361883 * A361885 A361886 A361887 KEYWORD nonn,easy AUTHOR Peter Bala, Mar 28 2023 STATUS approved

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Last modified June 19 19:59 EDT 2024. Contains 373507 sequences. (Running on oeis4.)
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The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A361884 a(n) = (1/n) * Sum_{k = 0..n} (-1)^(n+k) * (n + 2*k) * binomial(n+k-1,k)^3. 3
 2, 66, 2540, 110530, 5197752, 257490156, 13238524728, 699822144450, 37800431926400, 2077184897317816, 115757876008359312, 6526739641107783916, 371641758587326581200, 21341134886976332825400, 1234474507620634579565040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Compare with the closed form evaluation of the binomial sums (1/n) * Sum_{k = 0..n} (-1)^(n+k) * (n + 2*k) * binomial(n+k-1,k) = binomial(2*n,n) and (1/n) * Sum_{k = 0..n} (n + 2*k) * binomial(n+k-1,k)^2 = binomial(2*n,n)^2. The central binomial coefficients u(n) := binomial(2*n,n) = A000984(n) satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for positive integers n and r and all primes p >= 5. We conjecture that the present sequence satisfies the same congruences. More generally, for m >= 3, the sequences {b_m(n) : n >= 1} and {c_m(n) : n >= 1} defined by b_m(n) = (1/n) * Sum_{k = 0..n} (n + 2*k) * binomial(n+k-1,k)^m and c_m(n) = (1/n) * Sum_{k = 0..n} (-1)^k * (n + 2*k) * binomial(n+k-1,k)^m may both satisfy the same congruences. LINKS Table of n, a(n) for n=1..15. FORMULA a(n) ~ 2^(6*n) / (3 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 29 2023 MAPLE seq( (1/n)*add((-1)^(n+k) * (n + 2*k) * binomial(n+k-1, n-1)^3, k = 0..n), n = 1..20); MATHEMATICA Table[Sum[(-1)^(n+k) * (n + 2*k) * Binomial[n+k-1, k]^3, {k, 0, n}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Mar 29 2023 *) PROG (PARI) a(n) = (1/n) * sum(k = 0, n, (-1)^(n+k) * (n + 2*k) * binomial(n+k-1, k)^3); \\ Michel Marcus, Mar 30 2023 CROSSREFS Cf. A000984, A002894, A361883, A361885, A361886. Sequence in context: A218433 A092884 A230735 * A261784 A350730 A255435 Adjacent sequences: A361881 A361882 A361883 * A361885 A361886 A361887 KEYWORD nonn,easy AUTHOR Peter Bala, Mar 28 2023 STATUS approved

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Last modified June 19 19:59 EDT 2024. Contains 373507 sequences. (Running on oeis4.)
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The text passage is a detailed description of a sequence in the Online Encyclopedia of Integer Sequences (OEIS), including its definition, formula, and implementation in various programming languages. It does not contain any explicit mathematical questions or answers to such questions.

TYPE: NO_QUESTION