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

1,5

Comments

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...

Links

Table of n, a(n) for n=1..55.

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.

Formula

Zudilin gives a complicated general formula involving binomial coefficients, thus proving that all T(r, k) are integers.

Example

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, ...

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 *)

Prog

(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

Crossrefs

Rows 2-5 are A000172, A000658, A092868, A379610.

Columns 2-3 seem to be A000079, A081656.

Sequence in context: A213786 A055130 A051292 * A265241 A166888 A083677

Adjacent sequences: A094421 A094422 A094423 * A094425 A094426 A094427

Keyword

nonn,tabl,changed

Author

Ralf Stephan, May 16 2004

Extensions

More terms from R. J. Mathar, Oct 06 2006

Status

approved