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A246538 G.f.: Sum_{n>=0} 2^n * x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k]^2. 3
1, 3, 15, 87, 559, 3807, 26919, 195399, 1445967, 10859967, 82527687, 633165255, 4896345487, 38117454303, 298435452135, 2348094847047, 18554434810831, 147171478237695, 1171272947140359, 9349653181797063, 74834845484454927, 600441007306747167, 4828337830880795943, 38904633251921442375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) == 1 (mod 3) iff n = 4*A005836(k) for k>=0, and a(n) == 0 (mod 3) otherwise, where A005836 gives numbers n whose base 3 representation contains no 2.

LINKS

Table of n, a(n) for n=0..23.

FORMULA

G.f.: Sum_{n>=0} x^n / (1-2*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k]^2.

G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 2^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 * 2^(n-k) * Sum_{j=0..k} C(k,j)^2 * 2^j * x^j.

a(n) = Sum_{k=0..[n/2]} 2^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 2^j.

Recurrence: (n-4)*(n-1)^2*a(n) = 3*(n-4)*(3*n^2 - 9*n + 7)*a(n-1) - (n-2)*(n^2 - 6*n + 6)*a(n-2) - 3*(n-3)*(11*n^2 - 66*n + 92)*a(n-3) - 2*(n-4)*(n^2 - 6*n + 6)*a(n-4) + 12*(n-2)*(3*n^2 - 27*n + 61)*a(n-5) - 8*(n-5)^2*(n-2)*a(n-6). - Vaclav Kotesovec, Nov 05 2014

a(n) ~ ((3 + 4*sqrt(2) + sqrt(33+24*sqrt(2))))^n / (Pi *n * 2^(n+5/2)). - Vaclav Kotesovec, Nov 05 2014

EXAMPLE

G.f.: A(x) = 1 + 3*x + 15*x^2 + 87*x^3 + 559*x^4 + 3807*x^5 + 26919*x^6 +...

where

A(x) = 1/(1-x) + 2*x/(1-x)^3*(1+x)^2

+ 2^2*x^2/(1-x)^5*(1 + 2^2*x + x^2)^2

+ 2^3*x^3/(1-x)^7*(1 + 3^2*x + 3^2*x^2 + x^3)^2

+ 2^4*x^4/(1-x)^9*(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)^2

+ 2^5*x^5/(1-x)^11*(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)^2

+ 2^6*x^6/(1-x)^13*(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)^2 +...

MATHEMATICA

Table[Sum[2^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 2^j, {j, 0, n-2*k}], {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2014 *)

PROG

(PARI) /* By definition: */

{a(n)=local(A=1); A=sum(m=0, n, 2^m*x^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, 25, print1(a(n), ", "))

(PARI) /* By a binomial identity: */

{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 * 2^k * x^k)^2 +x*O(x^n)); polcoeff(A, n)}

for(n=0, 25, 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^k * sum(j=0, k, binomial(k, j)^2 * x^j)+x*O(x^n))), n)}

for(n=0, 25, 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 * 2^j * x^j)+x*O(x^n))), n)}

for(n=0, 25, print1(a(n), ", "))

(PARI) /* By a formula for a(n): */

{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 2^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 2^j))}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A246539, A246540, A246056, A246423, A243948, A245929, A227845, A245925, A005836.

Sequence in context: A278392 A168503 A089022 * A132371 A192989 A316666

Adjacent sequences:  A246535 A246536 A246537 * A246539 A246540 A246541

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 28 2014

STATUS

approved

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Last modified June 25 12:55 EDT 2019. Contains 324352 sequences. (Running on oeis4.)