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 A248055 G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k]. 0
 1, 5, 34, 265, 2219, 19490, 177119, 1651405, 15707416, 151791375, 1485814989, 14697965770, 146673432721, 1474490991635, 14915914368896, 151701887367585, 1550083118902041, 15903333300738320, 163749905809635219, 1691449817705302875, 17521670544878571584, 181972459046153912945 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * x^k]. G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 4^(k-j) * x^j. G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^j. a(n) = Sum_{k=0..[n/2]} Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j. a(n) ~ (11+3*sqrt(13))^(n+1) / (Pi * n * 2^(n+7/2)). - Vaclav Kotesovec, Oct 04 2014 EXAMPLE G.f.: A(x) = 1 + 5*x + 34*x^2 + 265*x^3 + 2219*x^4 + 19490*x^5 +... MATHEMATICA Table[Sum[Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 4^j, {j, 0, n-2*k}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 04 2014 *) PROG (PARI) /* By definition: */ {a(n, p=4, q=1)=local(A=1); A=sum(m=0, n, x^m/(1-p*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * q^k * x^k) * sum(k=0, m, binomial(m, k)^2 * p^k * x^k) +x*O(x^n)); polcoeff(A, n)} for(n=0, 25, print1(a(n, 4, 1), ", ")) (PARI) /* By a binomial identity: */ {a(n, p=4, q=1)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * p^(m-k) * q^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)} for(n=0, 25, print1(a(n, 4, 1), ", ")) (PARI) /* By a binomial identity: */ {a(n, p=4, q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * p^(k-j) * q^j * x^j)+x*O(x^n))), n)} for(n=0, 25, print1(a(n, 4, 1), ", ")) (PARI) /* By a binomial identity: */ {a(n, p=4, q=1)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * p^(m-k) * sum(j=0, k, binomial(k, j)^2 * q^j * x^j)+x*O(x^n))), n)} for(n=0, 25, print1(a(n, 4, 1), ", ")) (PARI) /* Formula for a(n): */ {a(n, p=4, q=1)=sum(k=0, n\2, sum(j=0, n-2*k, q^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * p^j))} for(n=0, 25, print1(a(n, 4, 1), ", ")) CROSSREFS Cf. A246510, A248053, A246423, A246455, A246056, A246813. Sequence in context: A083987 A002776 A081342 * A243659 A058248 A116435 Adjacent sequences:  A248052 A248053 A248054 * A248056 A248057 A248058 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 30 2014 STATUS approved

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Last modified December 15 17:42 EST 2019. Contains 330000 sequences. (Running on oeis4.)