A171426 Row sum of column n in A126441 or A161924.

1, 5, 16, 49, 129, 341, 833, 2029, 4760, 11068, 25182, 56888, 126661, 280169, 613893, 1337386, 2893793, 6232013, 13352607, 28497552, 60580905, 128368080, 271153740, 571224871, 1200298631, 2516483260, 5264785310, 10993631034, 22915508186, 47688470005

Offset: 1

Author

Alford Arnold, Dec 12 2009

Keywords

• easy
• nonn

Comments

From Emeric Deutsch, Sep 06 2017: (Start)

Row sums of the triangle A161919.

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)

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..60

Crossrefs

Cf. A161924, A171429.

Programs

Maple

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
# second Maple program:
b:= proc(n, i, l, r) option remember; `if`(n=0, r, `if`(i>n, 0,
      b(n, i+1, l, r)+b(n-i, i$2, ((x-> 2*x+1)@@(i-l))(2*r))))
    end:
a:= n-> b(n, 1, 0$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 01 2021

Mathematica

b[n_, i_, l_, r_] := b[n, i, l, r] = If[n == 0, r, If[i>n, 0,
     b[n, i+1, l, r] + b[n-i, i, i, Nest[2#+1&, 2r, i-l]]]];
a[n_] := b[n, 1, 0, 0];
Array[a, 30] (* Jean-François Alcover, Mar 02 2021, after Alois P. Heinz *)

Extensions

Ninth term corrected by Alford Arnold, Jan 22 2010

Offset changed to 1 and a(17)-a(27) from Emeric Deutsch, Sep 06 2017

a(28)-a(30) from Alois P. Heinz, Sep 07 2017