Search: id:A094424
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%I A094424
%S A094424 1,1,1,1,2,1,1,4,10,1,1,8,68,56,1,1,16,424,1732,346,1,1,32,2576,48896,
%T A094424 51076,2252,1,1,64,15520,1383568,6672232,1657904,15184,1,1,128,93248,
%U A094424 39776000,873960976,1022309408,57793316,104960,1,1,256,559744,1159151680,
               116758856608,615833930816,176808084544,2117525792,739162,1
%N A094424 Array read by antidiagonals: Solutions to Schmidt's Problem.
%C A094424 T(r,k) satisfies sum[k=0,n, C(n,k)^r*C(n+k,k)^r] = sum[k=0,n, C(n,k)*C(n+k,
               k)*T(r,k)] for all n=0,1,2,3...
%H A094424 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SchmidtsProblem.html">Schmidt's Problem</a>
%H A094424 W. Zudilin, <a href="http://www.combinatorics.org/Volume_11/Abstracts/
               v11i1r22.html">On a combinatorial problem of Asmus Schmidt</a>.
%F A094424 Zudilin gives a complicated general formula involving binomial coefficients, 
               thus proving that all T(r, k) are integers.
%e A094424 1 1 1 1 1 1
%e A094424 1 2 10 56 346 2252
%e A094424 1 4 68 1732 51076 1657904
%e A094424 1 8 424 48896 6672232 1022309408
%e A094424 1 16 2576 1383568 873960976 615833930816
%e A094424 1 32 15520 39776000 116758856608 371558588978432
%o A094424 (PARI) A094424row(r,kmax)={ local(nmat,rhs,cv) ; nmat=matrix(kmax+1,kmax+1) 
               ; rhs=matrix(kmax+1,1) ; for(n=0,kmax, for(k=0,kmax, nmat[n+1,k+1]=binomial(n,
               k)*binomial(n+k,k) ; ) ; rhs[n+1,1]=sum(i=0,n,binomial(n,i)^r*binomial(n+i,
               i)^r) ; ) ; cv=matsolve(nmat,rhs) ; } A094424(nmax)={ local(T,c) 
               ; T=matrix(nmax,nmax) ; for(r=1,nmax, c=A094424row(r,nmax-1) ; for(i=1,
               nmax, T[r,i]=c[i,1] ; ) ; ) ; return(T) ; } { rmax=10 ; T=A094424(rmax) 
               ; for(d=0,rmax-1, for(c=0,d, print1(T[d-c+1,c+1],",") ; ) ; ) ; } 
               - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 06 2006
%Y A094424 Rows 2-4 are A000172, A000658, A092868.
%Y A094424 Columns 2-3 seem to be A000079, A081656.
%Y A094424 Sequence in context: A111569 A055130 A051292 this_sequence A083677 A075803 
               A127966
%Y A094424 Adjacent sequences: A094421 A094422 A094423 this_sequence A094425 A094426 
               A094427
%K A094424 nonn,tabl
%O A094424 1,5
%A A094424 Ralf Stephan, May 16 2004
%E A094424 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 06 2006

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