A026933 Self-convolution of array T given by A008288.

1, 2, 11, 52, 269, 1414, 7575, 41064, 224665, 1237898, 6859555, 38187164, 213408805, 1196524814, 6727323439, 37915058384, 214140178225, 1211694546194, 6867622511675, 38981807403268, 221562006394173, 1260814207833750, 7182599953332423, 40958645048598840, 233779564099963081

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers. - Paul D. Hanna, Jan 10 2012

G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - Vladeta Jovovic, May 13 2003

a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - Benoit Cloitre, Sep 28 2005

G.f.: exp( Sum_{n>=1} A002203(n)^2/2 * x^n/n ), where A002203 are the companion Pell numbers. - Paul D. Hanna, Jan 10 2012

Self-convolution yields A204062; self-convolution of A204061. - Paul D. Hanna, Jan 10 2012

From Vaclav Kotesovec, Oct 08 2012: (Start)

Recurrence: n*a(n) = (5*n-3)*a(n-1) + (5*n-2)*a(n-2) - (n-1)*a(n-3).

a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). (End)

0 = +a(n)*(+a(n+1) -8*a(n+2) -7*a(n+3) +2*a(n+4)) +a(n+1)*(-2*a(n+1) +22*a(n+2) +20*a(n+3) -7*a(n+4)) +a(n+2)*(+30*a(n+2) +22*a(n+3) -8*a(n+4)) +a(n+3)*(-2*a(n+3) +a(n+4)) for all n in Z. - Michael Somos, Jun 27 2017

Mathematica

Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_]:= Sum[ SeriesCoefficient[ SeriesCoefficient[1/(1-x-y-x*y) , {x, 0, n-k}] , {y, 0, k}]^2, {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)
A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k, k-n, -n, -1])^2, {k, 0, n}];
Table[A026933[n], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

Prog

(PARI) /* Sum of squares of Delannoy numbers: */
{a(n)=sum(k=0, n, polcoeff(polcoeff(1/(1-x-y-x*y +x*O(x^n)+y*O(y^k)), n-k, x), k, y)^2)} \\ Paul D. Hanna, Jan 10 2012
(PARI) /* Involving squares of companion Pell numbers: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}
\\ Paul D. Hanna, Jan 10 2012
(PARI) my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ Joerg Arndt, May 04 2013
(Sage)
def A026933_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

Crossrefs

Cf. A002203, A008288, A204061, A204062.

Sequence in context: A026996 A110308 A027201 * A346925 A052171 A333542

Adjacent sequences: A026930 A026931 A026932 * A026934 A026935 A026936

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Vladeta Jovovic, May 13 2003

Status

approved