A137645 a(n) = Sum_{k=0..n} C((n-k)*k, k) * C((n-k)*k, n-k).

1, 0, 1, 4, 42, 608, 10986, 240492, 6167112, 181154848, 5995624710, 220711502648, 8943846698096, 395588177834784, 18962600075658460, 979198125493716492, 54189002212286942316, 3199366560075461850320, 200730550064907653703510, 13336507142191259122442532, 935401326531455246646557760, 69066745767857553528070539760, 5355032622687046432711489319940

Offset: 0

Author

Paul D. Hanna, Jan 31 2008

Keywords

• nonn

Examples

			The initial terms of this sequence are
a(0) = 1 = 1*1;
a(1) = 0 = 1*0 + 0*1;
a(2) = 1 = 1*0 + 1*1 + 0*1;
a(3) = 4 = 1*0 + 2*1 + 1*2 + 0*1;
a(4) = 42 = 1*0 + 3*1 + 6*6 + 1*3 + 0*1;
a(5) = 608 = 1*0 + 4*1 + 15*20 + 20*15 + 1*4 + 0*1;
a(6) = 10986 = 1*0 + 5*1 + 28*70 + 84*84 + 70*28 + 1*5 + 0*1;
a(7) = 240492 = 1*0 + 6*1 + 45*252 + 220*495 + 495*220 + 252*45 + 1*6 + 0*1; ...
where the triangle of coefficients binomial((n-k)*k, k) begins:
1;
1, 0;
1, 1, 0;
1, 2, 1, 0;
1, 3, 6, 1, 0;
1, 4, 15, 20, 1, 0;
1, 5, 28, 84, 70, 1, 0;
1, 6, 45, 220, 495, 252, 1, 0;
1, 7, 66, 455, 1820, 3003, 924, 1, 0;
1, 8, 91, 816, 4845, 15504, 18564, 3432, 1, 0;
1, 9, 120, 1330, 10626, 53130, 134596, 116280, 12870, 1, 0; ...
and the triangle A060539 of coefficients binomial((n-k)*k, n-k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 6, 3, 1;
0, 1, 20, 15, 4, 1;
0, 1, 70, 84, 28, 5, 1;
0, 1, 252, 495, 220, 45, 6, 1;
0, 1, 924, 3003, 1820, 455, 66, 7, 1;
0, 1, 3432, 18564, 15504, 4845, 816, 91, 8, 1;
0, 1, 12870, 116280, 134596, 53130, 10626, 1330, 120, 9, 1; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A060539.

Programs

PARI

{a(n)=sum(k=0,n,binomial((n-k)*k,k)*binomial((n-k)*k,n-k))}
for(n=0,25,print1(a(n),", "))