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A085812 Sum(sum(binomial(i,j),i=n..2*n),j=0..n). 4
1, 5, 22, 91, 366, 1454, 5748, 22691, 89590, 354010, 1400268, 5544334, 21973420, 87158972, 345977832, 1374249251, 5461704870, 21717305762, 86391846492, 343800647066, 1368639516420, 5450093895812, 21708897213912, 86492537630606 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

As the definition indicates, each term is the sum of numbers from Pascal's Triangle in an (n+1) X (n+1) square arrangement.

Example for a(2):

1

1  1

|1  2  1 |

|1  3  3 |1

|1  4  6 |4  1

1  5  10 10 5  1

1  6  15 20 15 6  1

...

and

(1+2+1) + (1+3+3) + (1+4+6) = 22 = a(2). Similarly, a(1) = (1+1)+(1+2) = 5. - John Molokach, Sep 17 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 4^n - 2^n + C(2*n+2, n+1)/2. - Vaclav Kotesovec, Oct 28 2012

(n+1)*a(n) +2*(-6*n-1)*a(n-1) +4*(13*n-9)*a(n-2) +8*(-12*n+19)*a(n-3) +32*(2*n-5)*a(n-4)=0. - R. J. Mathar, Oct 01 2013

EXAMPLE

a(1)=binomial(1,0)+binomial(2,0)+binomial(1,1)+binomial(2,1)=1+1+1+2=5

MAPLE

a := n->add(add(binomial(i, j), i=n..2*n), j=0..n); seq(a(n), n=0..25);

MATHEMATICA

Table[4^n-2^n+Binomial[2*n+2, n+1]/2, {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2012 *)

PROG

(PARI) a(n)=4^n-2^n+binomial(2*n+2, n+1)/2; \\ Joerg Arndt, May 10 2013

CROSSREFS

Sequence in context: A208736 A050185 A216597 * A172061 A211973 A053297

Adjacent sequences:  A085809 A085810 A085811 * A085813 A085814 A085815

KEYWORD

nonn,easy

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 25 2003

STATUS

approved

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Last modified January 18 13:09 EST 2019. Contains 319271 sequences. (Running on oeis4.)