A078812 -id:A078812 - cp's OEIS frontend

A206474 Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 0, 3, 3, 4, 4, 1, 1, 1, 1, 6, 6, 5, 5, 1, 1, 0, 4, 4, 10, 10, 6, 6, 1, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1

Offset: 0

Philippe Deléham, Feb 08 2012

Triangle T(n,k), read by rows, given by (1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Antidiagonal sums are A158780(n+1).
Row sums are 2*Fibonacci(n) = 2*A000045(n), n>0.

			Triangle begins :
1
1, 1
0, 1, 1
1, 1, 1, 1
0, 2, 2, 1, 1
1, 1, 3, 3, 1, 1
0, 3, 3, 4, 4, 1, 1
1, 1, 6, 6, 5, 5, 1, 1
0, 4, 4, 10, 10, 6, 6, 1, 1
1, 1, 10, 10, 15, 15, 7, 7, 1, 1
0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1

Cf. A007318, A078812, A085478, A128908,

t[1, 0] = 1; t[2, 0] = 0; t[n_, n_] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= n := t[n, k] = t[n-1, k-1] + t[n-2, k]; t[n_, k_] = 0; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)

T(2n, 2k) = A128908(n,k), T(2n+1, 2k) = T(2n+1, 2k+1) = A085478(n,k) = Binomial (n+k, 2k), T(2n+2, 2k+1) = A078812(n,k) = Binomial(n+k-1, 2k-1).
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(0,1) = 1, T(0,2) = 0.
G.f.: (1+x-x^2)/(1-x*y-x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A000129(n) (n>0), A000007(n), A135528(n-1), A055389(n) for x = -2, -1, 0, 1 respectively .

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

A115980 Array read by rows distributing the values of A000712 (vertically) and A001519 (horizontally).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 3, 2, 4, 6, 5, 4, 6, 2, 4, 6, 8, 6, 3, 4, 4, 8, 9, 2, 4, 6, 8, 10, 7, 4, 12, 8, 4, 8, 12, 12, 2, 4, 6, 8, 10, 12, 8, 3, 6, 5, 4, 14, 21, 12, 4, 8, 12, 16, 15, 2, 4, 6, 8, 10, 12, 14, 9

Offset: 0

Alford Arnold, Feb 11 2006

A001906 records the partial sums of the column sequence A001519 and is also the row sum of A078812 and of A085643; sequences linking a(n) to compositions of n having k parts when there are q kinds of part q. - Alford Arnold, Apr 30 2006

			The array begins:
1
..2
....2
....3
......2
......4..3
......4
.........2
.........4..4..3
.........6..6..4
.........5
with column sums beginning 1 2 5 10 20 ...A000712 related to A000041
and sums over each template beginning 1 2 5 13 34 ...A001519 related to A000045

Cf. A000041, A000045.

Cf. A001906, A078812, A085643.

cp's OEIS Frontend

A206474 Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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A115980 Array read by rows distributing the values of A000712 (vertically) and A001519 (horizontally).

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