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%I #16 Feb 21 2019 08:20:48
%S 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 51076,2252,1,1,64,15520,1383568,6672232,1657904,15184,1,1,128,93248,
%U 39776000,873960976,1022309408,57793316,104960,1,1,256,559744,1159151680,116758856608,615833930816,176808084544,2117525792,739162,1
%N Array read by antidiagonals: Solutions to Schmidt's Problem.
%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>
%H W. Zudilin, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i1r22.html">On a combinatorial problem of Asmus Schmidt</a>, Electron. J. Combin. 11:1 (2004), #R22, 8 pages.
%F Zudilin gives a complicated general formula involving binomial coefficients, thus proving that all T(r, k) are integers.
%e 1 1 1 1 1 1
%e 1 2 10 56 346 2252
%e 1 4 68 1732 51076 1657904
%e 1 8 424 48896 6672232 1022309408
%e 1 16 2576 1383568 873960976 615833930816
%e 1 32 15520 39776000 116758856608 371558588978432
%t eq[r_, n_] := eq[r, n] = Sum[Binomial[n, k]^r*Binomial[n + k, k]^r, {k, 0, n}] == Sum[Binomial[n, k]*Binomial[n + k, k]*t[r, k], {k, 0, n}]; c[r_, k_] := t[r, k] /. Solve[Table[eq[r, n], {n, 0, k}], t[r, k]] // First; lg = 10; m = Table[c[r, k], {r, 1, lg}, {k, 0, lg - 1}];
%t Flatten[ Table[ Reverse @ Diagonal[ Reverse /@ m, k],{k, lg - 1, -lg + 1, -1}]][[1 ;; 55]] (* _Jean-François Alcover_, Jul 20 2011 *)
%o (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_, Oct 06 2006
%Y Rows 2-4 are A000172, A000658, A092868.
%Y Columns 2-3 seem to be A000079, A081656.
%K nonn,tabl
%O 1,5
%A _Ralf Stephan_, May 16 2004
%E More terms from _R. J. Mathar_, Oct 06 2006
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