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Search: a003056 -id:a003056
Displaying 31-40 of 361 results found.
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A362209
Irregular triangle read by rows: T(n, k) is the number of k X k matrices using all the integers from 1 to k^2 and having trace equal to n, with 1 <= k <= A003056(n).
+20
3
1, 0, 0, 4, 0, 4, 0, 8, 0, 4, 4320, 0, 4, 4320, 0, 0, 8640, 0, 0, 12960, 0, 0, 17280, 11496038400, 0, 0, 21600, 11496038400, 0, 0, 30240, 22992076800, 0, 0, 30240, 34488115200, 0, 0, 34560, 57480192000, 0, 0, 34560, 68976230400, 291948240981196800000
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Table of n, a(n) for n=1..45.
FORMULA
T(n, k) = A362187(k)*A362208(n, k).
EXAMPLE
Irregular triangle begins:
1;
0;
0, 4;
0, 4;
0, 8;
0, 4, 4320;
0, 4, 4320;
0, 0, 8640;
0, 0, 12960;
0, 0, 17280, 11496038400;
0, 0, 21600, 11496038400;
0, 0, 30240, 22992076800;
0, 0, 30240, 34488115200;
0, 0, 34560, 57480192000;
0, 0, 34560, 68976230400, 291948240981196800000;
...
T(5,2) = 8 since we have:
[1, 2] [1, 3] [4, 2] [4, 3]
[3, 4], [2, 4], [3, 1], [2, 1],
.
[2, 1] [2, 4] [3, 1] [3, 4]
[4, 3], [1, 3], [4, 2], [1, 2].
MATHEMATICA
A362208[n_, k_] := Length[Select[Join@@Permutations/@Select[IntegerPartitions[n, All, Range[k^2]], UnsameQ@@#&], Length[#]==k&]]; Table[(k^2-k)!A362208[n, k], {n, 15}, {k, Floor[(Sqrt[8n+1]-1)/2]}]//Flatten
CROSSREFS
Cf. A000290, A003056 (row lengths), A345132, A362187, A362208.
KEYWORD
nonn,tabf
AUTHOR
Stefano Spezia, Apr 11 2023
STATUS
approved
A050601
Recursion counts for summation table A003056 with formula a(0,x) = x, a(y,0) = y, a(y,x) = a((y XOR x),2*(y AND x))
+20
2
0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 1, 2, 2, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 1, 3, 3, 2, 2, 3, 3, 1, 0, 0, 2, 1, 3, 2, 1, 2, 3, 1, 2, 0, 0, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 3, 2, 3, 1, 3, 1, 3, 1, 3, 2, 3, 0, 0, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 0, 0, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 0
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,12
LINKS
Table of n, a(n) for n=0..119.
FORMULA
a(n) -> add2c( (n-((trinv(n)*(trinv(n)-1))/2)), (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) )
MAPLE
add2c := proc(a, b) option remember; if((0 = a) or (0 = b)) then RETURN(0); else RETURN(1+add_c(XORnos(a, b), 2*ANDnos(a, b))); fi; end;
MATHEMATICA
trinv[n_] := Floor[(1/2)*(Sqrt[8*n + 1] + 1)];
Sum2c[a_, b_] := Sum2c[a, b] = If[0 == a || 0 == b, Return[0], Return[ Sum2c[BitXor[a, b], 2*BitAnd[a, b]] + 1]];
a[n_] := Sum2c[n - (1/2)*trinv[n]*(trinv[n] - 1), (trinv[n] - 1)*(trinv[ n]/2 + 1) - n];
Table[a[n], {n, 0, 120}](* Jean-François Alcover, Mar 07 2016, adapted from Maple *)
CROSSREFS
Cf. A050600, A050602, A003056, A048720 (for the Maple implementation of trinv and XORnos, ANDnos)
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Jun 22 1999
STATUS
approved
A166021
a(n) = 2*A000124(A003056(n-1)) if A002262(n-1)=0, otherwise a(n-1)+1.
+20
2
2, 4, 5, 8, 9, 10, 14, 15, 16, 17, 22, 23, 24, 25, 26, 32, 33, 34, 35, 36, 37, 44, 45, 46, 47, 48, 49, 50, 58, 59, 60, 61, 62, 63, 64, 65, 74, 75, 76, 77, 78, 79, 80, 81, 82, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..65.
PROG
(MIT Scheme:) (define (A166021 n) (if (zero? (A002262 (-1+ n))) (* 2 (A000124 (A003056 (-1+ n)))) (1+ (A166021 (-1+ n)))))
CROSSREFS
Complement of A136272.
KEYWORD
nonn,tabl,easy
AUTHOR
Antti Karttunen, Oct 05 2009
STATUS
approved
A291955
Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 1, -1, 0, -1, 0, 0, 1, 0, 0, 2, -1, 0, 0, -2, -1, 1, 0, 0, 1, 1, -1, -1, 0, -1, -3, 2, 0, 0, 3, 2, -1, -1, 1, 0, -2, -3, 2, 1, 0, 0, 3, 4, -3, -1, 0, 0, -4, -4, 3, 2, -1, 0, 4
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,38
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, 1;
0, 0;
0, 0, 1;
0, -1, 0;
0, 0, 0;
0, 0, -1, 1;
0, 1, 0, 0;
0, 0, -1, 0;
0, 1, 1, -1;
0, -1, 0, 0, 1.
CROSSREFS
Row sums give A291956.
Columns 0-1 give A000007, A049346.
Cf. A291904.
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 06 2017
STATUS
approved
A291957
Triangle read by rows: T(n,k) = T(n-k,k-1) - T(n-k,k) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, 1, 0, -1, 0, 1, 1, 0, 0, -1, 0, -1, 0, 0, 1, 1, 1, 0, 0, -1, -1, 0, -1, 1, 0, 0, 2, 0, 0, 0, -2, -2, 0, 1, 0, 0, 2, 1, -1, 0, 2, 0, 0, 0, 0, -2, -1, -1, 0, 0, 1, 2, 0, -1, 0, 1, -2, 0, 2, 1, 0, -3, -1, 0, 0, -1, 0, 2, 3, 1, -1, 0, 0, 1, -2, 0, 1, 0, 0, -3, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,28
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, 1;
0, -1;
0, 1, 1;
0, 0, -1;
0, -1, 0;
0, 1, 1, 1;
0, 0, -1, -1;
0, -1, 1, 0;
0, 2, 0, 0;
0, -2, -2, 0, 1.
CROSSREFS
Row sums give A291958.
Columns 0-1 give A000007, (-1)*A111165 (for n>0).
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 06 2017
STATUS
approved
A291969
Triangle read by rows: T(n,k) = k * (T(n-k,k-1) + T(n-k,k)) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 0, 1, 6, 6, 0, 1, 14, 6, 0, 1, 14, 18, 0, 1, 30, 36, 0, 1, 30, 60, 24, 0, 1, 62, 96, 24, 0, 1, 62, 198, 72, 0, 1, 126, 270, 144, 0, 1, 126, 474, 336, 0, 1, 254, 780, 480, 120, 0, 1, 254, 1188, 1080, 120, 0, 1, 510, 1800
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
FORMULA
G.f. of column k: k! * x^(k*(k+1)/2) / Product_{j=1..k} (1-j*x^j).
EXAMPLE
First few rows are:
1;
0, 1;
0, 1;
0, 1, 2;
0, 1, 2;
0, 1, 6;
0, 1, 6, 6;
0, 1, 14, 6;
0, 1, 14, 18;
0, 1, 30, 36;
0, 1, 30, 60, 24.
CROSSREFS
Row sums give A204858.
Columns 0-1 give A000007, A000012.
Cf. A291960.
KEYWORD
nonn,tabf
AUTHOR
Seiichi Manyama, Sep 07 2017
STATUS
approved
A292130
Triangle read by rows: T(n,k) = (-3) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, -3, 0, -3, 0, -3, 9, 0, -3, 9, 0, -3, 18, 0, -3, 18, -27, 0, -3, 27, -27, 0, -3, 27, -54, 0, -3, 36, -81, 0, -3, 36, -108, 81, 0, -3, 45, -135, 81, 0, -3, 45, -189, 162, 0, -3, 54, -216, 243, 0, -3, 54, -270, 405, 0, -3, 63, -324, 486, -243, 0, -3, 63, -378
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, -3;
0, -3;
0, -3, 9;
0, -3, 9;
0, -3, 18;
0, -3, 18, -27;
0, -3, 27, -27;
0, -3, 27, -54;
0, -3, 36, -81;
0, -3, 36, -108, 81.
CROSSREFS
Row sums give A292128.
Columns 0-1 give A000007, (-1)*A010701.
Cf. A291971.
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 09 2017
STATUS
approved
A292142
Triangle read by rows: T(n,k) = (-4)^(k mod 2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, -4, 0, -4, 0, -4, -4, 0, -4, -4, 0, -4, -8, 0, -4, -8, 16, 0, -4, -12, 16, 0, -4, -12, 32, 0, -4, -16, 48, 0, -4, -16, 64, 16, 0, -4, -20, 80, 16, 0, -4, -20, 112, 32, 0, -4, -24, 128, 48, 0, -4, -24, 160, 80, 0, -4, -28, 192, 96, -64, 0, -4, -28, 224, 144
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, -4;
0, -4;
0, -4, -4;
0, -4, -4;
0, -4, -8;
0, -4, -8, 16;
0, -4, -12, 16;
0, -4, -12, 32;
0, -4, -16, 48;
0, -4, -16, 64, 16.
CROSSREFS
Row sums give A292135.
Columns 0-1 give A000007, (-4)*A000012.
Cf. A292047.
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 09 2017
STATUS
approved
A292143
Triangle read by rows: T(n,k) = (-4)^(1 - k mod 2) * T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).
+20
2
1, 0, 1, 0, 1, 0, 1, -4, 0, 1, -4, 0, 1, -8, 0, 1, -8, -4, 0, 1, -12, -4, 0, 1, -12, -8, 0, 1, -16, -12, 0, 1, -16, -16, 16, 0, 1, -20, -20, 16, 0, 1, -20, -28, 32, 0, 1, -24, -32, 48, 0, 1, -24, -40, 80, 0, 1, -28, -48, 96, 16, 0, 1, -28, -56, 144, 16, 0, 1, -32
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
0, 1;
0, 1;
0, 1, -4;
0, 1, -4;
0, 1, -8;
0, 1, -8, -4;
0, 1, -12, -4;
0, 1, -12, -8;
0, 1, -16, -12;
0, 1, -16, -16, 16.
CROSSREFS
Row sums give A292141.
Columns 0-1 give A000007, A000012.
Cf. A292049.
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 09 2017
STATUS
approved
A292781
Triangle read by rows: T(n,k) = T(n-k,k-1) with T(0,0) = 1 and T(n,0) = -1/n * Sum_{k=1..A003056(n)} (-1)^k * (2*k+1) * (n+1-A060544(k+1)) * T(n,k).
+20
2
1, -24, 1, 252, -24, -1472, 252, 1, 4830, -1472, -24, -6048, 4830, 252, -16744, -6048, -1472, 1, 84480, -16744, 4830, -24, -113643, 84480, -6048, 252, -115920, -113643, -16744, -1472, 534612, -115920, 84480, 4830, 1, -370944, 534612, -113643, -6048, -24
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Seiichi Manyama, Rows n = 0..481, flattened
EXAMPLE
First few rows are:
1;
-24, 1;
252, -24;
-1472, 252, 1;
4830, -1472, -24;
-6048, 4830, 252;
-16744, -6048, -1472, 1;
84480, -16744, 4830, -24;
-113643, 84480, -6048, 252;
-115920, -113643, -16744, -1472;
534612, -115920, 84480, 4830, 1.
-----------------------------------------
n=5
T(5,1) = T(4,0) = 4830, T(5,2) = T(3,1) = 252.
T(5,0) = -1/5 * Sum_{k=1..2} (-1)^k * (2*k+1) * (5+1-A060544(k+1)) * T(n,k) = -1/5 * ((-3)*(-4)*4830 + 5*(-22)*252) = -6048.
n=6
T(6,1) = T(5,0) = -6048, T(6,2) = T(4,1) = -1472, T(6,3) = T(3,2) = 1.
T(6,0) = -1/6 * Sum_{k=1..3} (-1)^k * (2*k+1) * (6+1-A060544(k+1)) * T(n,k) = -1/6 * ((-3)*(-3)*(-6048) + 5*(-21)*(-1472) - 7*(-48)*1) = -16744.
PROG
(Ruby)
def A292781(n)
ary = [[1]]
(1..n).each{|i|
m = ((Math.sqrt(1 + 8 * i) - 1) / 2).to_i
a = (1..m).map{|j| ary[i - j][j - 1]}
ary << [-(1..m).inject(0){|s, j| s + (-1) ** (j % 2) * (2 * j + 1) * (i - 9 * j * (j + 1) / 2) * a[j - 1]} / i] + a
}
ary.flatten
end
p A292781(20)
CROSSREFS
All columns are A000594.
Cf. A003056, A060544.
KEYWORD
sign,tabf
AUTHOR
Seiichi Manyama, Sep 23 2017
STATUS
approved
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