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 A246455 G.f.: Sum_{n>=0} x^n / (1-2*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k]. 6
 1, 3, 14, 75, 439, 2718, 17483, 115575, 779716, 5342937, 37065029, 259694334, 1834494025, 13048232853, 93352065596, 671245767795, 4847769647701, 35145881098272, 255674861620439, 1865618971630305, 13650358024120804, 100123485522075447, 736036670999528711 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 2^(k-j) * x^j. G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 2^(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, k)^2 * 2^j. Recurrence: (n-5)*(n-4)*(n-2)*n^2*a(n) = 3*(n-5)*(n-4)*(4*n^3 - 12*n^2 + 10*n - 3)*a(n-1) - (n-5)*(n-4)*(n-1)*(34*n^2 - 102*n + 77)*a(n-2) - 3*(n-5)*(n-2)*(2*n-5)*a(n-3) + (n-3)*(53*n^4 - 636*n^3 + 2617*n^2 - 4254*n + 2226)*a(n-4) - 3*(n-4)*(n-1)*(2*n-7)*a(n-5) - (n-5)*(n-2)*(n-1)*(34*n^2 - 306*n + 689)*a(n-6) + 3*(n-2)*(n-1)*(4*n^3 - 60*n^2 + 298*n - 489)*a(n-7) - (n-6)^2*(n-4)*(n-2)*(n-1)*a(n-8). - Vaclav Kotesovec, Aug 27 2014 a(n) ~ c * d^n / n, where d = 5/2 + sqrt(2) + sqrt(29+20*sqrt(2))/2 = 7.698532234454699125... is the root of the equation d^4 - 10*d^3 + 19*d^2 - 10*d + 1 = 0, and c = 0.51515814369432019442929480754730417101213524575391074352368... . - Vaclav Kotesovec, Aug 27 2014 EXAMPLE G.f.: A(x) = 1 + 3*x + 14*x^2 + 75*x^3 + 439*x^4 + 2718*x^5 + 17483*x^6 +... where the g.f. is given by the binomial series: A(x) = 1/(1-2*x) + x/(1-2*x)^3 * (1+x) * (1+2*x) + x^2/(1-2*x)^5 * (1 + 2^2*x + x^2) * (1 + 2^2*2*x + 4*x^2) + x^3/(1-2*x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3) * (1 + 3^2*2*x + 3^2*4*x^2 + 8*x^3) + x^4/(1-2*x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4) * (1 + 4^2*2*x + 6^2*4*x^2 + 4^2*8*x^3 + 16*x^4) + x^5/(1-2*x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5) * (1 + 5^2*2*x + 10^2*4*x^2 + 10^2*8*x^3 + 5^2*16*x^4 + 32*x^5) +... We can also express the g.f. by the binomial series: A(x) = 1 + x*(1 + (2+x)) + x^2*(1 + 2^2*(2+x) + (4+2^2*2*x+x^2)) + x^3*(1 + 3^2*(2+x) + 3^2*(4+2^2*2*x+x^2) + (8+3^2*4*x+3^2*2*x^2+x^3)) + x^4*(1 + 4^2*(2+x) + 6^2*(4+2^2*2*x+x^2) + 4^2*(8+3^2*4*x+3^2*2*x^2+x^3) + (16+4^2*8*x+6^2*4*x^2+4^2*2*x^3+x^4)) + x^5*(1 + 5^2*(2+x) + 10^2*(4+2^2*2*x+x^2) + 10^2*(8+3^2*4*x+3^2*2*x^2+x^3) + 5^2*(16+4^2*8*x+6^2*4*x^2+4^2*2*x^3+x^4) + (32+5^2*16*x+10^2*8*x^2+10^2*4*x^3+5^2*2*x^4+x^5)) +... MATHEMATICA Table[Sum[Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, k]^2 * 2^j, {j, 0, n-2*k}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2014 *) PROG (PARI) /* By definition: */ {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-2*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2*x^k) * sum(k=0, m, binomial(m, k)^2*2^k*x^k) +x*O(x^n)); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* By a binomial identity: */ {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*sum(j=0, k, binomial(k, j)^2*2^(k-j)*x^j)+x*O(x^n))), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* By a binomial identity: */ {a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 2^(m-k) * sum(j=0, k, binomial(k, j)^2*x^j)+x*O(x^n))), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* Formula for a(n): */ {a(n)=sum(k=0, n\2, sum(j=0, n-2*k, binomial(n-k, k+j)^2*binomial(k+j, k)^2*2^j))} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A243948 (dual), A246538. Sequence in context: A026004 A200718 A063016 * A133798 A100937 A223026 Adjacent sequences:  A246452 A246453 A246454 * A246456 A246457 A246458 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 26 2014 EXTENSIONS Minor edits by Vaclav Kotesovec, Nov 05 2014 STATUS approved

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Last modified October 23 01:24 EDT 2018. Contains 316518 sequences. (Running on oeis4.)