A246563 G.f.: Sum_{n>=0} x^n / (1-x^2)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^(2*k)]^2.
1, 1, 2, 6, 15, 39, 116, 340, 1001, 3041, 9322, 28718, 89363, 279987, 881376, 2788464, 8860677, 28256709, 90407666, 290124182, 933482527, 3010689527, 9731366060, 31516942060, 102259648701, 332347297141, 1081810639970, 3526399820374, 11510355762339, 37616896717155
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 39*x^5 + 116*x^6 + 340*x^7 +... where the g.f. is given by the binomial series identity: A(x) = 1/(1-x^2) + x/(1-x^2)^3 * (1 + x^2)^2 + x^2/(1-x^2)^5 * (1 + 2^2*x^2 + x^4)^2 + x^3/(1-x^2)^7 * (1 + 3^2*x^2 + 3^2*x^4 + x^6)^2 + x^4/(1-x^2)^9 * (1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)^2 + x^5/(1-x^2)^11 * (1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10)^2 +... equals the series A(x) = 1/(1-x) + x^2/(1-x)^3 * (1 + x)^2 + x^4/(1-x)^5 * (1 + 2^2*x + x^2)^2 + x^6/(1-x)^7 * (1 + 3^2*x + 3^2*x^2 + x^3)^2 + x^8/(1-x)^9 * (1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2 + x^10/(1-x)^11 * (1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2 +... Curiously, the BISECTIONS of the g.f. are squares of integer series: let A(x) = B0(x^2) + x*B1(x^2), then B0(x) = 1 + 2*x + 15*x^2 + 116*x^3 + 1001*x^4 + 9322*x^5 + 89363*x^6 +...+ A246570(n)*x^n +... sqrt(B0(x)) = 1 + x + 7*x^2 + 51*x^3 + 425*x^4 + 3879*x^5 + 36527*x^6 +...+ A246572(n)*x^n +... B1(x) = 1 + 6*x + 39*x^2 + 340*x^3 + 3041*x^4 + 28718*x^5 + 279987*x^6 +...+ A246571(n)*x^n +... sqrt(B1(x)) = 1 + 3*x + 15*x^2 + 125*x^3 + 1033*x^4 + 9385*x^5 + 88531*x^6 +...+ A246573(n)*x^n +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..500
Programs
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Mathematica
Table[Sum[Sum[Binomial[n-k-j,k]^2 * Binomial[k,j]^2,{j,0,k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Sep 02 2014 *) -
PARI
/* By definition: */ {a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x^2)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^(2*k))^2 +x*O(x^n)); polcoeff(A, n)} for(n=0, 35, print1(a(n), ", ")) -
PARI
/* By a binomial identity: */ {a(n)=local(A=1); A=sum(m=0, n, x^(2*m)/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * x^k)^2 +x*O(x^n)); polcoeff(A, n)} for(n=0, 35, 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 * x^k * sum(j=0, k, binomial(k, j)^2 * x^j )+x*O(x^n))), n)} for(n=0, 35, 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 * x^(m-k) * sum(j=0, k, binomial(k, j)^2 * x^(2*j) )+x*O(x^n))), n)} for(n=0, 35, print1(a(n), ", ")) -
PARI
/* From a formula for a(n): */ {a(n)=sum(k=0, n\2, sum(j=0, min(k,n-2*k), binomial(n-k-j, k)^2 * binomial(k, j)^2 ))} for(n=0, 35, print1(a(n), ", "))
Formula
G.f.: Sum_{n>=0} x^(2*n) / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k * Sum_{j=0..k} C(k,j)^2 * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^(n-k) * Sum_{j=0..k} C(k,j)^2 * x^(2*j).
a(n) = Sum_{k=0..[n/2]} Sum_{j=0..k} C(n-k-j,k)^2 * C(k,j)^2.
Recurrence: n^2*(2*n - 7)*(2*n - 5)*a(n) = (2*n - 7)*(4*n^3 - 14*n^2 + 10*n - 3)*a(n-1) + (8*n^4 - 64*n^3 + 174*n^2 - 198*n + 87)*a(n-2) + (2*n - 3)*(20*n^3 - 150*n^2 + 362*n - 281)*a(n-3) + 3*(2*n - 5)*a(n-4) - (2*n - 7)*(20*n^3 - 150*n^2 + 362*n - 279)*a(n-5) - (8*n^4 - 96*n^3 + 414*n^2 - 742*n + 447)*a(n-6) - (2*n - 3)*(4*n^3 - 46*n^2 + 170*n - 197)*a(n-7) + (n-5)^2*(2*n - 5)*(2*n - 3)*a(n-8). - Vaclav Kotesovec, Sep 02 2014
a(n) ~ c * d^n / (Pi*n), where d = ((54+6*sqrt(33))^(2/3) + 12 + 3*(54+6*sqrt(33))^(1/3)) / (3*(54+6*sqrt(33))^(1/3)) = 3.3829757679062374941227... is the root of the equation -1 - d - 3*d^2 + d^3 = 0, c = 1/12*(199+3*sqrt(33))^(1/3) + 17/(6*(199+3*sqrt(33))^(1/3)) + 7/12 = 1.55556563078009965666864... . - Vaclav Kotesovec, Sep 02 2014
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