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 A008608 Number of n X n upper triangular matrices A of nonnegative integers such that a_1i + a_2i + ... + a_{i-1,i} - a_ii - a_{i,i+1} - ... - a_in = -1. 12
 1, 2, 7, 40, 357, 4820, 96030, 2766572, 113300265, 6499477726, 515564231770, 55908184737696, 8203615387086224, 1613808957720017838, 422045413500096791377, 145606442599303799948900, 65801956684134601408784992, 38698135339344702725297294600, 29437141738828506134939056167071, 28800381656420765181010517468370560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Garsia and Haglund call these Tesler matrices. - N. J. A. Sloane, Jul 04 2014 This is also the value of the type A_n Kostant partition function evaluated at (1,1,...,1,-n) in ZZ^(n+1). This is the number of ways of writing the vector (1,1,...,1,-n) in ZZ^(n+1) as a linear combination with nonnegative integer coefficients of the vectors e_i - e_j, for 1 <= imul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n): a := n -> multcoeff(n+1, F(n+1), [seq(1, i=1..n), -n], n+2): seq(a(i), i=2..7) # Alejandro H. Morales, Mar 11 2014, Jun 28 2015 # second Maple program: b:= proc(n, i, l) option remember; (m-> `if`(m=0, 1, `if`(i=0, b(l[1]+1, m-1, subsop(1=NULL, l)), add( b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n))))(nops(l)) end: a:= n-> b(1, n-1, [0\$(n-1)]): seq(a(n), n=1..14); # Alois P. Heinz, Jul 05 2015 MATHEMATICA b[n_, i_, l_List] := b[n, i, l] = Function[{m}, If[m==0, 1, If[i==0, b[l[[1]]+1, m-1, ReplacePart[l, 1 -> Sequence[]]], Sum[b[n-j, i-1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]][Length[l]]; a[n_] := b[1, n-1, Array[0&, n-1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jul 16 2015, after Alois P. Heinz *) CROSSREFS Cf. A259485, A259666. Row sums of A259786. Main diagonal (shifted) of A259841. Column k=1 of A259844. Sequence in context: A224677 A064626 A137731 * A028441 A006455 A130715 Adjacent sequences: A008605 A008606 A008607 * A008609 A008610 A008611 KEYWORD nonn AUTHOR Glenn P. Tesler (gptesler(AT)euclid.ucsd.edu) EXTENSIONS a(7)-a(13) from Alejandro H. Morales, Mar 12 2014 a(14) from Alejandro H. Morales, Jun 04 2015 a(15)-a(22) from Alois P. Heinz, Jul 05 2015 a(23) from Jay Pantone, Nov 19 2016 STATUS approved

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Last modified December 6 02:53 EST 2022. Contains 358594 sequences. (Running on oeis4.)