A097939
Sum of the smallest parts of all compositions of n.
Original entry on oeis.org
1, 3, 6, 12, 22, 42, 79, 151, 291, 566, 1106, 2175, 4293, 8499, 16864, 33523, 66727, 132958, 265137, 529050, 1056169, 2109282, 4213710, 8419697, 16827079, 33634489, 67237513, 134424624, 268768414, 537407062, 1074605619, 2148875961, 4297212424, 8593556211, 17185713097, 34369170909
Offset: 1
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A097939:=n->add(add(binomial(n-r*(k+1)-1,k), k=0..floor((n-r-1)/(r+1))), r=0..n-1): seq(A097939(n), n=1..50); # Wesley Ivan Hurt, Dec 03 2016
# second Maple Program:
b:= proc(n, m) option remember; `if`(n=0, m,
add(b(n-j, min(j, m)), j=1..n))
end:
a:= n-> b(n$2):
seq(a(n), n=1..40); # Alois P. Heinz, Jul 26 2020
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Drop[ CoefficientList[ Series[ Sum[x^k/(1 - x - x^k), {k, 50}], {x, 0, 35}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)
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N=66; x='x+O('x^N);
gf= sum(k=1,N, x^k/(1-x-x^k) );
Vec(gf)
/* Joerg Arndt, Jan 01 2013 */
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{a(n)=polcoeff(sum(m=1,n,x^m*sumdiv(m,d,1/(1-x +x*O(x^n))^d) ),n)}
A099233
Square array read by antidiagonals associated to sections of 1/(1-x-x^k).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 8, 1, 1, 1, 6, 15, 26, 28, 13, 1, 1, 1, 7, 21, 45, 69, 60, 21, 1, 1, 1, 8, 28, 71, 140, 181, 129, 34, 1, 1, 1, 9, 36, 105, 251, 431, 476, 277, 55, 1, 1, 1, 10, 45, 148, 413, 882, 1326, 1252, 595, 89, 1
Offset: 0
Rows begin
1, 1, 1, 1, 1, 1, ...
1, 1, 2, 3, 5, 8, ...
1, 1, 3, 6, 13, 28, ...
1, 1, 4, 10, 26, 69, ...
1, 1, 5, 15, 45, 140, ...
Row 1 is the 0-section of 1/(1-x-x) (A000079);
Row 2 is the 1-section of 1/(1-x-x^2) (A000045);
Row 3 is the 2-section of 1/(1-x-x^3) (A000930);
Row 4 is the 3-section of 1/(1-x-x^4) (A003269);
etc.
A125104
Triangle read by rows counting compositions (ordered partitions) by minimal part size.
Original entry on oeis.org
1, 1, 1, 1, 0, 3, 1, 0, 1, 6, 1, 0, 0, 2, 13, 1, 0, 0, 1, 3, 27, 1, 0, 0, 0, 2, 5, 56, 1, 0, 0, 0, 1, 2, 9, 115, 1, 0, 0, 0, 0, 2, 3, 15, 235, 1, 0, 0, 0, 0, 1, 2, 5, 25, 478, 1, 0, 0, 0, 0, 0, 2, 2, 8, 42, 969, 1, 0, 0, 0, 0, 0, 1, 2, 3, 12, 70, 1959, 1, 0, 0, 0, 0, 0, 0, 2, 2, 5, 18, 116, 3952, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 8, 27, 192, 7959, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 11, 41, 317, 16007
Offset: 0
Row 4 of the array is (1, 0, 1, 6) because there are six compositions with minimum part of size one: 1111, 31, 13, 211, 121, 112; one of size two: 22; none of size three; and 1 of size four: 4.
Triangle (after 45-degree counterclockwise rotation) begins:
1 1 3 6 13 27 56 115 235 478 969 1959 3952 7959
.1 0 1 2 3 5 9 15 25 42 70 116 192
..1 0 0 1 2 2 3 5 8 12 18 27
...1 0 0 0 1 2 2 2 3 5 8
....1 0 0 0 0 1 2 2 2 2
.....1 0 0 0 0 0 1 2 2
......1 0 0 0 0 0 0 1
.......1 0 0 0 0 0 0
........1 0 0 0 0 0
A144409
Antidiagonal expansion of: f(t,n) = If[n == 1, 1/(1 - t), 1/(1 - t^floor(n/2) - t^n)].
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 2, 5, 1, 1, 0, 1, 0, 3, 8, 1, 1, 0, 0, 0, 2, 4, 13, 1, 1, 0, 0, 1, 1, 0, 6, 21, 1, 1, 0, 0, 1, 0, 1, 3, 9, 34, 1, 1, 0, 0, 0, 0, 0, 1, 0, 13, 55, 1, 1, 0, 0, 0, 1, 0, 2, 2, 5, 19, 89, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 28, 144, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3
Offset: 1
{1},
{1, 1},
{1, 1, 1},
{1, 1, 2, 1},
{1, 0, 1, 3, 1},
{1, 0, 1, 2, 5, 1},
{1, 0, 1, 0, 3, 8, 1},
{1, 0, 0, 0, 2, 4, 13, 1},
{1, 0, 0, 1, 1, 0, 6, 21, 1},
{1, 0, 0, 1, 0, 1, 3, 9, 34, 1},
{1, 0, 0, 0, 0, 0, 1, 0, 13, 55, 1},
{1, 0, 0, 0, 1, 0, 2, 2, 5, 19, 89, 1},
{1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 28, 144, 1},
{1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 8, 41, 233, 1},
{1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 60, 377, 1}
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f[t_, n_] = If[n == 1, 1/(1 - t), 1/(1 - t^Floor[n/2] - t^n)]; a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}] ; Flatten[b]
Showing 1-4 of 4 results.
Comments