A094424 Array read by antidiagonals: Solutions to Schmidt's Problem.
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, 51076, 2252, 1, 1, 64, 15520, 1383568, 6672232, 1657904, 15184, 1, 1, 128, 93248, 39776000, 873960976, 1022309408, 57793316, 104960, 1, 1, 256, 559744, 1159151680, 116758856608, 615833930816, 176808084544, 2117525792, 739162, 1
Offset: 1
Examples
Array begins: 1, 1, 1, 1, 1, 1, ... 1, 2, 10, 56, 346, 2252, ... 1, 4, 68, 1732, 51076, 1657904, ... 1, 8, 424, 48896, 6672232, 1022309408, ... 1, 16, 2576, 1383568, 873960976, 615833930816, ... 1, 32, 15520, 39776000, 116758856608, 371558588978432, ...
Links
- Eric Weisstein's World of Mathematics, Schmidt's Problem
- W. Zudilin, On a combinatorial problem of Asmus Schmidt, Electron. J. Combin. 11:1 (2004), #R22, 8 pages.
Programs
-
Mathematica
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}]; Flatten[ Table[ Reverse @ Diagonal[ Reverse /@ m, k],{k, lg - 1, -lg + 1, -1}]][[1 ;; 55]] (* Jean-François Alcover, Jul 20 2011 *) -
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
Formula
Zudilin gives a complicated general formula involving binomial coefficients, thus proving that all T(r, k) are integers.
Extensions
More terms from R. J. Mathar, Oct 06 2006
Comments