A154948 -id:A154948 - cp's OEIS frontend

A154949 Diagonal sums of Riordan array A154948.

1, 1, 3, 5, 10, 18, 34, 62, 115, 211, 389, 715, 1316, 2420, 4452, 8188, 15061, 27701, 50951, 93713, 172366, 317030, 583110, 1072506, 1972647, 3628263, 6673417, 12274327, 22576008, 41523752, 76374088, 140473848, 258371689, 475219625

Offset: 0

Paul Barry, Jan 17 2009

Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1,-1)

a=0; b=0; c=0; lst={}; Do[z=a+b+c+1; AppendTo[lst,z]; a=b; b=c; c=z; z=a+b+c; AppendTo[lst,z]; a=b; b=c; c=z,{n,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 17 2010 *)
LinearRecurrence[{1,2,0,-1,-1},{1,1,3,5,10},40] (* Harvey P. Dale, Nov 13 2022 *)

G.f.: 1/((1-x^2)(1 - x - x^2 - x^3)).
a(n) = sum{k=0..floor(n/2), sum{j=0..n-k+1, C(n-k+1-j,k+1)C(k-1,j)}}.
a(n) = -A000035(n)/2 + A001590(n+4)/2. - R. J. Mathar, Oct 25 2012

A048776 First partial sums of A048739; second partial sums of A000129.

Original entry on oeis.org

1, 4, 12, 32, 81, 200, 488, 1184, 2865, 6924, 16724, 40384, 97505, 235408, 568336, 1372096, 3312545, 7997204, 19306972, 46611168, 112529329, 271669848, 655869048, 1583407968, 3822685009, 9228778012, 22280241060, 53789260160, 129858761409, 313506783008

Offset: 0

Barry E. Williams

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A001333, A000129, A048739.

with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008

a=b=0;Table[c=2*b+a+n;a=b;b=c,{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)
LinearRecurrence[{4,-4,0,1},{1,4,12,32},30] (* Harvey P. Dale, Aug 27 2014 *)

a(n) = 2*a(n-1) + a(n-2) + n + 1; a(0)=1, a(1)=4.
a(n) = (((7/2 + (5/2)*sqrt(2))*(1+sqrt(2))^n - (7/2 - (5/2)*sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2)) - (n+3)/2.
a(n) = (A000129(n+3) - (n+3))/2 = Sum_{j} A047662(n-j+1, j+1). - Henry Bottomley, Jul 09 2001
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)
Define an array with m(n,1)=1 and m(1,k) = k*(k+1)/2 for n=1,2,3,...  The interior terms are m(n,k) = m(n,k-1) + m(n-1,k-1) + m(n-1,k). The sum of the terms in each antidiagonal=a(n). - J. M. Bergot, Dec 01 2012 [This is A154948 without the first column. The diagonal is m(n,n) = A161731(n-1). R. J. Mathar, Dec 06 2012]
E.g.f.: exp(x)*(10*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - 2*(3 + x))/4. - Stefano Spezia, May 13 2023

More terms from Harvey P. Dale, Aug 27 2014

A154950 Riordan array (1/(1-x^4), x(1+x)/(1+x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 2, 1, 1, -1, -1, 3, 1, 0, 2, -4, 0, 4, 1, 0, 2, 2, -8, 2, 5, 1, 0, -2, 8, -2, -12, 5, 6, 1, 1, -2, -2, 18, -12, -15, 9, 7, 1, 0, 3, -12, 8, 28, -29, -16, 14, 8, 1, 0, 3, 3, -32, 38, 31, -53, -14, 20, 9, 1

Offset: 0

Paul Barry, Jan 17 2009

Row sums are A008619. Diagonal sums are A103221. Equal to A154948 times inverse of A007318.

			Triangle begins
1,
0, 1,
0, 1, 1,
0, -1, 2, 1,
1, -1, -1, 3, 1,
0, 2, -4, 0, 4, 1,
0, 2, 2, -8, 2, 5, 1,
0, -2, 8, -2, -12, 5, 6, 1,
1, -2, -2, 18, -12, -15, 9, 7, 1

Triangle T(n,k)=sum{i=0..n, sum{j=0..n+1, C(n+1-j,i+1)*C(i-1,j)}*(-1)^(i-k)*C(i,k)}.

cp's OEIS Frontend

A154949 Diagonal sums of Riordan array A154948.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A048776 First partial sums of A048739; second partial sums of A000129.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A154950 Riordan array (1/(1-x^4), x(1+x)/(1+x^2)).

Views

Author

Keywords

Comments

Examples

Formula