A154948 -id:A154948

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A154949

Diagonal sums of Riordan array A154948.

+20
1

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	LINKS	`Table of n, a(n) for n=0..33.` `Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1,-1)`
	FORMULA	`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`
	MATHEMATICA	`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 *)`
	KEYWORD	`easy,nonn`
	AUTHOR	`Paul Barry, Jan 17 2009`
	STATUS	`approved`

A048776

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

+10
9

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Table of n, a(n) for n=0..29.` `Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).`
	FORMULA	`a(n) = 2a(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)/2sqrt(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) = 4a(n-1) - 4a(n-2) + a(n-4).` `G.f.: -1/((x^2+2x-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)(10cosh(sqrt(2)x) + 7sqrt(2)sinh(sqrt(2)x) - 2(3 + x))/4. - Stefano Spezia, May 13 2023`
	MAPLE	`with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008`
	MATHEMATICA	`a=b=0; Table[c=2b+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 *)`
	CROSSREFS	`Cf. A001333, A000129, A048739.`
	KEYWORD	`easy,nonn`
	AUTHOR	`Barry E. Williams`
	EXTENSIONS	`More terms from Harvey P. Dale, Aug 27 2014`
	STATUS	`approved`

A154950

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

+10
1

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 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,9`
	COMMENTS	`Row sums are A008619. Diagonal sums are A103221. Equal to A154948 times inverse of A007318.`
	LINKS	`Table of n, a(n) for n=0..65.`
	FORMULA	`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)}.`
	EXAMPLE	`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`
	KEYWORD	`easy,sign,tabl`
	AUTHOR	`Paul Barry, Jan 17 2009`
	STATUS	`approved`

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