A026933 - OEIS

%I #40 May 26 2021 02:15:32

%S 1,2,11,52,269,1414,7575,41064,224665,1237898,6859555,38187164,

%T 213408805,1196524814,6727323439,37915058384,214140178225,

%U 1211694546194,6867622511675,38981807403268,221562006394173,1260814207833750,7182599953332423,40958645048598840,233779564099963081

%N Self-convolution of array T given by A008288.

%H Vincenzo Librandi, <a href="/A026933/b026933.txt">Table of n, a(n) for n = 0..200</a>

%F 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

%F G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - _Vladeta Jovovic_, May 13 2003

%F a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - _Benoit Cloitre_, Sep 28 2005

%F 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

%F Self-convolution yields A204062; self-convolution of A204061. - _Paul D. Hanna_, Jan 10 2012

%F From _Vaclav Kotesovec_, Oct 08 2012: (Start)

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

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

%F 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

%t Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)

%t 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 *)

%t A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k,k-n,-n,-1])^2, {k,0,n}];

%t Table[A026933[n], {n, 0, 40}] (* _G. C. Greubel_, May 25 2021 *)

%o (PARI) /* Sum of squares of Delannoy numbers: */

%o {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

%o (PARI) /* Involving squares of companion Pell numbers: */

%o {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

%o {a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}

%o \\ _Paul D. Hanna_, Jan 10 2012

%o (PARI) my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ _Joerg Arndt_, May 04 2013

%o (Sage)

%o def A026933_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()

%o A026933_list(40) # _G. C. Greubel_, May 25 2021

%Y Cf. A002203, A008288, A204061, A204062.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E More terms from _Vladeta Jovovic_, May 13 2003