A322518 Binomial transform of the Apéry numbers (A005259).
1, 6, 84, 1680, 39240, 999216, 26899896, 752939424, 21691531800, 638947312080, 19155738105504, 582589712312064, 17930566188602136, 557417298916695600, 17477836958370383280, 552090876791399769600, 17552554240486710112920, 561230779055361080132880
Offset: 0
Examples
a(2) = binomial(2,0)*A(0) + binomial(2,1)*A(1) + binomial(2,2)*A(2), where A(k) denotes the k-th Apéry number. Using this definition: a(2) = binomial(2,0)*(binomial(0,0)*binomial(0,0))^2 + binomial(2,1)*((binomial(1,0)*binomial(1,0))^2 + (binomial(1,1)*binomial(2,1))^2) + binomial(2,2)*((binomial(2,0)*binomial(2,0))^2 + (binomial(2,1)*binomial(3,1))^2 + (binomial(2,2)*binomial(4,2))^2) = 84.
Links
- Jackson Earles, Justin Ford, Poramate Nakkirt, Marlo Terr, Dr. Ilia Mishev, Sarah Arpin, Binomial Transforms of Sequences, Fall 2018.
- N. J. A. Sloane, Transforms
Programs
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Julia
function BinomialTransform(seq) N = length(seq) bt = Array{BigInt,1}(undef,N) bt[1] = seq[1] for k in 1:N-1 next = BigInt(0) for j in 0:k next += binomial(k, j)*seq[j+1] end bt[k+1] = next end bt end BinomialTransform([A005259(n) for n in 0:18]) |> println # Peter Luschny, Jan 06 2020 -
Mathematica
a[n_] := Sum[Binomial[n, k] * Sum[(Binomial[k, j] * Binomial[k+j, j])^2, {j, 0, k}], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *) -
Sage
def OEISbinomial_transform(N, seq): BT = [seq[0]] k = 1 while k< N: next = 0 j = 0 while j <=k: next = next + ((binomial(k,j))*seq[j]) j = j+1 BT.append(next) k = k+1 return BT Apery = oeis('A005259') OEISBinom = OEISbinomial_transform(18,Apery.first_terms(20))
Formula
a(n) ~ 2^(n - 3/4) * 3^(n + 3/2) * (1 + sqrt(2))^(2*n - 1) / (Pi*n)^(3/2). - Vaclav Kotesovec, Dec 17 2018
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all primes p and n a positive integer. - Peter Bala, Jan 06 2020
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