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

Author

Clark Kimberling

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Crossrefs

Cf. A002203, A008288, A204061, A204062.

Programs

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 *)

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. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

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

Extensions

More terms from Vladeta Jovovic, May 13 2003