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A249891 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 * (-x)^k]. 2
1, -1, -11, 59, 241, -3361, -419, 172451, -575399, -7443481, 58900909, 216416771, -4229184959, 2577683471, 244556409709, -1057605544621, -11063790445559, 106391247022391, 291287267857021, -7745138594921101, 10707792104722681, 449533741924068119, -2373288757544551451 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The signs of the terms form a seemingly unpredictable pattern.

The sequence begins [1, 0, -1, 0, -11, 0, 59, 0, ...]. As is standard practice in the database, the zeros have been omitted.

Even order terms of the sequence (i.e., the nonzero ones) gives the diagonal of rational function 1/(1 - (x + y + x*z - y*z - x*y*z)). - Gheorghe Coserea, Aug 29 2018

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

G.f.: Sum_{n>=0} (-x)^n * Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j =  1 - x^2 - 11*x^4 + 59*x^6 + 241*x^8 - ....

a(n) = Sum_{k=0..n} (-1)^k * Sum_{j=0..2*n-2*k} C(2*n-k, k+j)^2 * C(k+j, j)^2.

From Peter Bala, Mar 13 2018: (Start)

a(n) = Sum_{k = 0..n} (-1)^k*C(2*k,k)*C(n+k,n-k)^2.

n^2*(2*n-5)*a(n) = -( (2*n-5)*(2*n-1)^2*a(n-1) + (2*n-3)*(70*n^2-210*n+87)*a(n-2) + (2*n-1)*(2*n-5)^2*a(n-3) + (2*n-1)*(n-3)^2*a(n-4) ). (End)

a(n) = hypergeom([-n, -n, n + 1, n + 1], [1/2, 1, 1], -1/4). - Peter Luschny, Mar 14 2018

y = Sum_{n>=0} a(n)*x^n satisfies: 0 = 2*x*(x^4 + 4*x^3 + 70*x^2 + 4*x + 1)*y''' - (3*x^6 + 12*x^5 + 190*x^4 - 52*x^3 - 837*x^2 - 32*x - 4)*y'' - (9*x^5 + 24*x^4 + 166*x^3 - 106*x^2 - 947*x - 18)*y' - (3*x^4 + 3*x^3 - 67*x^2 - 29*x - 106)*y. - Gheorghe Coserea, Aug 29 2018

EXAMPLE

G.f.: A(x) = 1 - x^2 - 11*x^4 + 59*x^6 + 241*x^8 - 3361*x^10 - 419*x^12 + ...

where the g.f. is given by the series:

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

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

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

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

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

in which the coefficients of the odd powers of x vanish.

We can also express the g.f. by the series:

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

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

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

- x^5*(1 - 5^2*(1+x) + 10^2*(1+2^2*x+x^2) - 10^2*(1+3^2*x+3^2*x^2+x^3) + 5^2*(1+4^2*x+6^2*x^2+4^2*x^3+x^4) - (1+5^2*x+10^2*x^2+10^2*x^3+5^2*x^4+x^5)) + ...

MAPLE

seq(add((-1)^k*binomial(2*k, k)*binomial(n+k, n-k)^2, k = 0..n), n = 0..20); # Peter Bala, Mar 13 2018

MATHEMATICA

a[n_] := HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1/2, 1, 1}, -1/4]; Table[a[n], {n, 0, 22}] (* Peter Luschny, Mar 14 2018 *)

PROG

(PARI) /* By definition: */

{a(n)=local(A=1); A=sum(m=0, n, x^m/(1+x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 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(2*n), ", "))

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

{a(n)=polcoeff(sum(m=0, n, (-x)^m*sum(k=0, m, (-1)^k * binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * x^j) +x*O(x^n))), n)}

for(n=0, 25, print1(a(2*n), ", "))

(PARI) /* Formula for a(2*n): */

{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, (-1)^j * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 ))}

for(n=0, 25, print1(a(2*n), ", "))

(PARI)

seq(N) = {

  my(a=vector(N), t1, t2); a[1]=-1; a[2]=-11; a[3]=59; a[4]=241;

  for (n=5, N,

    t1 = (2*n-5)*(2*n-1)^2*a[n-1] + (2*n-3)*(70*n^2-210*n+87)*a[n-2];

    t2 = (2*n-1)*(2*n-5)^2*a[n-3] + (2*n-1)*(n-3)^2*a[n-4];

    a[n] = -(t1 + t2)/(n^2*(2*n-5)));

  concat(1, a);

};

seq(22)

\\ test: y=Ser(seq(303), 'x); 0 == 2*x*(x^4 + 4*x^3 + 70*x^2 + 4*x + 1)*y''' - (3*x^6 + 12*x^5 + 190*x^4 - 52*x^3 - 837*x^2 - 32*x - 4)*y'' - (9*x^5 + 24*x^4 + 166*x^3 - 106*x^2 - 947*x - 18)*y' - (3*x^4 + 3*x^3 - 67*x^2 - 29*x - 106)*y

\\ Gheorghe Coserea, Aug 29 2018

CROSSREFS

Cf. A245925, A246056.

Sequence in context: A142401 A139859 A217114 * A186256 A164299 A253207

Adjacent sequences:  A249888 A249889 A249890 * A249892 A249893 A249894

KEYWORD

sign,easy

AUTHOR

Paul D. Hanna, Nov 07 2014

STATUS

approved

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Last modified August 8 11:31 EDT 2020. Contains 336298 sequences. (Running on oeis4.)