A058393
A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0
Offset: 0
Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.
Rows are
A000035 (
A000012 with zeros),
A000012,
A040000 etc. Columns are
A000012,
A001477,
A000124,
A000125,
A000127,
A006261,
A008859,
A008860,
A008861,
A008862,
A008863 etc. Diagonals include
A000079,
A000225,
A000295,
A002662,
A002663,
A002664,
A035038,
A035039,
A035040,
A035041, etc. The triangles
A008949,
A054143 and
A055248 also appear in the half of the array which is not powers of 2.
A058395
Square array read by antidiagonals. Based on triangular numbers (A000217) with each term being the sum of 2 consecutive terms in the previous row.
Original entry on oeis.org
1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 6, 3, 4, 3, 1, 0, 6, 6, 6, 4, 1, 10, 6, 9, 10, 9, 5, 1, 0, 10, 12, 15, 16, 13, 6, 1, 15, 10, 16, 21, 25, 25, 18, 7, 1, 0, 15, 20, 28, 36, 41, 38, 24, 8, 1, 21, 15, 25, 36, 49, 61, 66, 56, 31, 9, 1, 0, 21, 30, 45, 64, 85, 102, 104, 80, 39, 10, 1, 28, 21, 36, 55, 81, 113, 146, 168, 160, 111, 48, 11, 1
Offset: 0
The array T(n, k) starts:
[0] 1, 0, 3, 0, 6, 0, 10, 0, 15, 0, ...
[1] 1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...
[2] 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...
[3] 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
[4] 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
[5] 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, ...
[6] 1, 6, 18, 38, 66, 102, 146, 198, 258, 326, ...
[7] 1, 7, 24, 56, 104, 168, 248, 344, 456, 584, ...
[8] 1, 8, 31, 80, 160, 272, 416, 592, 800, 1040, ...
[9] 1, 9, 39, 111, 240, 432, 688, 1008, 1392, 1840, ...
The triangle
A055252 also appears in half of the array.
-
gf := n -> (1 + x)^n / (1 - x^2)^3: ser := n -> series(gf(n), x, 20):
seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..9)), n = 0..9); # Peter Luschny, Apr 12 2023
-
T[0, k_] := If[OddQ[k], 0, (k+2)(k+4)/8];
T[n_, k_] := T[n, k] = If[k == 0, 1, T[n-1, k-1] + T[n-1, k]];
Table[T[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 13 2023 *)
A062534
Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).
Original entry on oeis.org
1, -2, 1, 3, -1, 1, -4, 2, 0, 1, 5, -2, 2, 1, 1, -6, 3, 0, 3, 2, 1, 7, -3, 3, 3, 5, 3, 1, -8, 4, 0, 6, 8, 8, 4, 1, 9, -4, 4, 6, 14, 16, 12, 5, 1, -10, 5, 0, 10, 20, 30, 28, 17, 6, 1, 11, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1, -12, 6, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 13, -6, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, -14, 7, 0, 21, 70, 175
Offset: 0
Showing 1-3 of 3 results.
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