The On-Line Encyclopedia of Integer Sequences (OEIS)

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A108296

Diagonal sums of the number triangle associated to A086617.
(history; published version)

#10 by Alois P. Heinz at Mon Sep 02 21:57:00 EDT 2024

STATUS

editing

approved

#9 by Alois P. Heinz at Mon Sep 02 21:56:57 EDT 2024

FORMULA

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} binomial(n-2*k, j)*binomial(k, j) * A000108(j).

STATUS

approved

editing

#8 by Alois P. Heinz at Mon Sep 02 21:56:43 EDT 2024

STATUS

proposed

approved

#7 by Jason Yuen at Mon Sep 02 21:52:40 EDT 2024

STATUS

editing

proposed

#6 by Jason Yuen at Mon Sep 02 21:52:21 EDT 2024

FORMULA

G.f.: (1-x^2-sqrt(1-2x2*x^2-4x4*x^3-3x3*x^4))/(2*x^3*(1-x^2)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-2k, j)C(k, j)C(j)}.

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2*k} binomial(n-2*k, j)*binomial(k, j)*A000108(j).

STATUS

approved

editing

#5 by R. J. Mathar at Fri Nov 16 04:10:11 EST 2012

STATUS

editing

approved

#4 by R. J. Mathar at Fri Nov 16 04:10:01 EST 2012

FORMULA

Conjecture: (n+3)*a(n) +(-n-2)*a(n-1) +2*(-n-1)*a(n-2) +2*(-n+3)*a(n-3) +(n+1)*a(n-4) +3*(n-2)*a(n-5)=0. - R. J. Mathar, Nov 16 2012

STATUS

approved

editing

#3 by Russ Cox at Fri Mar 30 18:59:08 EDT 2012

AUTHOR

_Paul Barry (pbarry(AT)wit.ie), _, May 31 2005

Discussion

Fri Mar 30

18:59

OEIS Server: https://oeis.org/edit/global/287

#2 by N. J. A. Sloane at Fri Feb 24 03:00:00 EST 2006

FORMULA

G.f.: (1-x^2-sqrt(1-2x^2-4x^3-3x^4))/(2*x^3(1-x^2)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-2k, j)C(k, j)C(j)}.

KEYWORD

easy,nonn,new

#1 by N. J. A. Sloane at Tue Jul 19 03:00:00 EDT 2005

NAME

Diagonal sums of the number triangle associated to A086617.

DATA

1, 1, 2, 3, 5, 8, 14, 24, 43, 78, 144, 269, 509, 971, 1868, 3618, 7049, 13805, 27162, 53661, 106405, 211697, 422458, 845386, 1696017, 3410522, 6873060, 13878721, 28077439, 56900936, 115501012, 234807488, 478032437, 974507543, 1989123814

OFFSET

0,3

COMMENTS

The triangle associated to A086617 is given by T(n,k)=if(k<=n, sum{j=0..n-k, C(n-k,j)C(k,j)C(j)},0). A050253(n)=A108296(n+2)-A108296(n).

FORMULA

G.f.: (1-x^2-sqrt(1-2x^2-4x^3-3x^4))/(2*x^3(1-x^2)); a(n)=sum{k=0..floor(n/2), sum{j=0..n-2k, C(n-2k,j)C(k,j)C(j)}.

KEYWORD

easy,nonn

AUTHOR

Paul Barry (pbarry(AT)wit.ie), May 31 2005

STATUS

approved