A171426 - OEIS

%I #40 Mar 02 2021 03:31:35

%S 1,5,16,49,129,341,833,2029,4760,11068,25182,56888,126661,280169,

%T 613893,1337386,2893793,6232013,13352607,28497552,60580905,128368080,

%U 271153740,571224871,1200298631,2516483260,5264785310,10993631034,22915508186,47688470005

%N Row sum of column n in A126441 or A161924.

%C From _Emeric Deutsch_, Sep 06 2017: (Start)

%C Row sums of the triangle A161919.

%C a(n) = sum of the viabin numbers of the partitions of n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [3,1,1] of 5. The southeast border of its Ferrers board yields 10011, leading to the viabin number 19. (End)

%H Alois P. Heinz, <a href="/A171426/b171426.txt">Table of n, a(n) for n = 1..60</a>

%e The values 8,9,6,11,15 map to 1111,211,22,31,4, respectively; so a(4) = 8+9+6+11+15 = 49.

%p with(combinat): a := proc (n) local ff, partovi: ff := proc (X) local s: s := [1, seq(0, j = 1 .. X[2])]: s := map(convert, s, string): return cat(op(s)) end proc: partovi := proc (P) local X, n, Y, i: X := convert(P, multiset): n := X[-1][1]: Y := map(proc (t) options operator, arrow: t[1] end proc, X): for i to n do if member(i, Y) = false then X := [op(X), [i, 0]] end if end do: X := sort(X, proc (s, t) options operator, arrow: evalb(s[1] < t[1]) end proc): X := map(ff, X); X := cat(op(X)): n := parse(X): n := convert(n, decimal, binary): (1/2)*n end proc: add(partovi(partition(n)[j]), j = 1 .. numbpart(n)) end proc: seq(a(n), n = 1 .. 27); # the subprogram partovi (due to W. Edwin Clark) yields the viabin number of a given partition. # _Emeric Deutsch_, Sep 06 2017

%p # second Maple program:

%p b:= proc(n, i, l, r) option remember; `if`(n=0, r, `if`(i>n, 0,

%p       b(n, i+1, l, r)+b(n-i, i$2, ((x-> 2*x+1)@@(i-l))(2*r))))

%p     end:

%p a:= n-> b(n, 1, 0$2):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Mar 01 2021

%t b[n_, i_, l_, r_] := b[n, i, l, r] = If[n == 0, r, If[i>n, 0,

%t      b[n, i+1, l, r] + b[n-i, i, i, Nest[2#+1&, 2r, i-l]]]];

%t a[n_] := b[n, 1, 0, 0];

%t Array[a, 30] (* _Jean-François Alcover_, Mar 02 2021, after _Alois P. Heinz_ *)

%Y Cf. A161924, A171429.

%K easy,nonn

%O 1,2

%A _Alford Arnold_, Dec 12 2009

%E Ninth term corrected by _Alford Arnold_, Jan 22 2010

%E Offset changed to 1 and a(17)-a(27) from _Emeric Deutsch_, Sep 06 2017

%E a(28)-a(30) from _Alois P. Heinz_, Sep 07 2017