A115086 -id:A115086 - cp's OEIS frontend

A115081 Column 0 of triangle A115080.

1, 1, 3, 11, 50, 257, 1467, 9081, 60272, 424514, 3151226, 24510411, 198870388, 1676878231, 14648843341, 132228263355, 1230505582380, 11782173683640, 115878367974480, 1168833058344870, 12075008262774120, 127608480923659770

Offset: 0

Paul D. Hanna, Jan 13 2006, Nov 19 2006

nonn

Also equals row sums of triangle A125080.

			At n=5, a(5) = Sum_{k=0..2} A000108(5-k)*A001147(k)*C(5,2*k) so that a(5) = 42*1*C(5,0) + 14*1*C(5,2) + 5*3*C(5,4) = 42*1*1 + 14*1*10 + 5*3*5 = 42 + 140 + 75 = 257.

Cf. A115080, A115082 (column 1), A115083 (column 2), A115084 (row sums); A115086.

Cf. A125080 (related triangle); A000108, A001147.

{a(n)=sum(k=0,n\2,binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)*k!/2^k*binomial(n,2*k))}

{a(n)=sum(k=0,n\2,(2*n-2*k)!*n!/k!/(n-k)!/(n-k+1)!/(n-2*k)!/2^k )}

a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A001147(k)*C(n,2*k), where A000108 is the Catalan numbers and A001147 is the double factorials.

a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A000108(k)*(k+1)!*C(n,2k)/2^k where A000108(n) = C(2n,n)/(n+1) are the Catalan numbers. a(n) = Sum_{k=0..n} (-1)^(n-k)*n!/k!*A115082(k) . - Paul D. Hanna, Feb 19 2007

A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 5, 3, 1, 58, 21, 7, 4, 1, 321, 102, 32, 9, 5, 1, 1963, 579, 158, 45, 11, 6, 1, 13053, 3601, 933, 226, 60, 13, 7, 1, 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1, 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1, 5599204, 1359906, 306070

Offset: 0

Paul D. Hanna, Jan 13 2006

Triangle A115080 is the dual of this triangle.

			T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:
12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;
21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;
102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;
158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.
Triangle begins:
1;
1, 1;
3, 2, 1;
12, 5, 3, 1;
58, 21, 7, 4, 1;
321, 102, 32, 9, 5, 1;
1963, 579, 158, 45, 11, 6, 1;
13053, 3601, 933, 226, 60, 13, 7, 1;
92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;
702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;
5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;
46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;
407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..405, as a flattened triangle of rows 0..27.

Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).

{T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n-1,j+k)*T(j+k+1,k+1))))}
for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

A115087 Column 1 of triangle A115085.

Original entry on oeis.org

1, 2, 5, 21, 102, 579, 3601, 24426, 176858, 1359906, 10996740, 93136545, 821734072, 7529402477, 71404315785, 699210393923, 7053151126260, 73158494370744, 778922147312076, 8500606925203992, 94961268583354364, 1084632255967292304

Offset: 0

Paul D. Hanna, Jan 13 2006

nonn

Cf. A115085, A115086 (column 0), A115088 (column 2), A115089 (row sums); A115082.

A115088 Column 2 of triangle A115085.

Original entry on oeis.org

1, 3, 7, 32, 158, 933, 5939, 41385, 306070, 2403792, 19799468, 170676213, 1530176012, 14235081941, 136908376331, 1358591876186, 13876456179864, 145643421890800, 1568070473632644, 17295212337571836, 195161233968831068

Offset: 0

Paul D. Hanna, Jan 13 2006

nonn

Cf. A115085, A115086 (column 0), A115087 (column 1), A115089 (row sums); A115083.

A115089 Row sums of triangle A115085.

Original entry on oeis.org

1, 2, 6, 21, 91, 470, 2763, 17894, 125113, 932702, 7347025, 60761449, 524852444, 4716259252, 43936770258, 423178496553, 4203717419747, 42980494150963, 451494062943969, 4865135046178557, 53701937703205383, 606454041389250791

Offset: 0

Paul D. Hanna, Jan 13 2006

nonn

Cf. A115085, A115086 (column 0), A115087 (column 1), A115088 (column 2); A115084.

cp's OEIS Frontend

A115081 Column 0 of triangle A115080.

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A115087 Column 1 of triangle A115085.

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A115088 Column 2 of triangle A115085.

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A115089 Row sums of triangle A115085.

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