%I A030662
%S A030662 1,5,19,69,251,923,3431,12869,48619,184755,705431,2704155,
%T A030662 10400599,40116599,155117519,601080389,2333606219,9075135299,
%U A030662 35345263799,137846528819,538257874439,2104098963719,8233430727599
%N A030662 Number of combinations of n things from 1 to n at a time, with repeats 
               allowed.
%C A030662 Comment from Ralf Stephan, May 17 2004: Add terms of an increasingly 
               bigger diamond-shaped part of Pascal's triangle:
%C A030662 .......................... 1
%C A030662 ............ 1 .......... 1 1
%C A030662 .. 1 ...... 1 1 ........ 1 2 1
%C A030662 . 1 1 =5 . 1 2 1 =19 .. 1 3 3 1 =69
%C A030662 .. 2 ...... 3 3 ........ 4 6 4
%C A030662 ............ 6 ......... 10 10
%C A030662 .......................... 20
%C A030662 The prime p divides a((p-1)/2) for p = 5,13,17,29,37,41,53,61,73,89,97.. 
               = A002144[n] Pythagorean primes: primes of form 4n+1. - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%C A030662 Also, number of square submatrices of a square matrix. - Jono Henshaw 
               (jjono(AT)hotmail.com), Apr 22 2008
%F A030662 2*(2*n-1)!/(n!*(n-1)!)-1.
%F A030662 a(n) = Sum_{k=1..n} binomial(n, k)^2. - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Aug 20 2002
%F A030662 a(n) = Sum_{j=0..n} Sum_{i=j..n+j} binomial(i, j). - Klaus Strassburger 
               (strass(AT)ddfi.uni-duesseldorf.de), Jul 23 2003
%F A030662 a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1} binomial(i+j, i). - N. J. A. Sloane, 
               Jan 31 2009
%F A030662 Also for n>1: a(n)=(2*n)!/(n!)^2-1 - Hugo Pfoertner (hugo(AT)pfoertner.org), 
               Feb 10 2004
%F A030662 a(n) = Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}]. - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%F A030662 Equals A115112 + 1. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
%p A030662 seq(sum((binomial(n,m))^2,m=1..n),n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 19 2008
%p A030662 f:=n->add( add( binomial(i+j,i), i=0..n),j=0..n); [seq(f(n),n=0..12)]; 
               - N. J. A. Sloane, Jan 31 2009
%t A030662 Table[Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}],{n,1,20}] - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006
%Y A030662 2*A001700 - 1. Cf. A047909, A091908, A144660, A002144.
%Y A030662 Equals A000984 - 1. Central column of triangle A014473.
%Y A030662 Right-hand column 2 of triangle A102541.
%Y A030662 Sequence in context: A143954 A047145 A055991 this_sequence A149758 A026590 
               A095073
%Y A030662 Adjacent sequences: A030659 A030660 A030661 this_sequence A030663 A030664 
               A030665
%K A030662 nonn,nice
%O A030662 1,2
%A A030662 Donald Mintz (djmintz(AT)home.com)