A165253 -id:A165253 - cp's OEIS frontend

A186024 Inverse of eigentriangle of triangle A085478.

1, -1, 1, -1, -1, 1, -1, -3, -1, 1, -1, -6, -5, -1, 1, -1, -10, -15, -7, -1, 1, -1, -15, -35, -28, -9, -1, 1, -1, -21, -70, -84, -45, -11, -1, 1, -1, -28, -126, -210, -165, -66, -13, -1, 1, -1, -36, -210, -462, -495, -286, -91, -15, -1, 1, -1, -45, -330, -924, -1287, -1001, -455, -120, -17, -1, 1

Offset: 0

Paul Barry, Feb 10 2011

Row sums are A186025.

			Triangle begins
1,
-1, 1,
-1, -1, 1,
-1, -3, -1, 1,
-1, -6, -5, -1, 1,
-1, -10, -15, -7, -1, 1,
-1, -15, -35, -28, -9, -1, 1,
-1, -21, -70, -84, -45, -11, -1, 1,
-1, -28, -126, -210, -165, -66, -13, -1, 1,
-1, -36, -210, -462, -495, -286, -91, -15, -1, 1,
-1, -45, -330, -924, -1287, -1001, -455, -120, -17, -1, 1

Cf. A129818, A109954, A220670, A123970, A085478, A165253.

T(n,k)=if(k

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

A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 4, 1, 6, 5, 1, 4, 10, 6, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 36, 210, 462, 495, 286, 91, 15, 1

Offset: 0

Roger L. Bagula, May 01 2010

Row sums are in A052952.

Contains the right half of each row of A030528. - R. J. Mathar, May 19 2013

			1;
1;
2, 1;
3, 1;
3, 4, 1;
6, 5, 1;
4, 10, 6, 1;
10, 15, 7, 1;
5, 20, 21, 8, 1;
15, 35, 28, 9, 1;
6, 35, 56, 36, 10, 1;
21, 70, 84, 45, 11, 1;
7, 56, 126, 120, 55, 12, 1;
28, 126, 210, 165, 66, 13, 1;
8, 84, 252, 330, 220, 78, 14, 1;
36, 210, 462, 495, 286, 91, 15, 1;

Cf. A180987 (read diagonally downwards), A098925, A026729, A085478, A165253

t[n_, m_] := Binomial[m + 1, n - m];
Table[Table[t[n, m], {m, Floor[(n + 1)/2], n}], {n, 0, 15}];
Flatten[%]

T(m,n)=binomial(n+1,m-n) \\ Charles R Greathouse IV, May 19 2013

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A186024 Inverse of eigentriangle of triangle A085478.

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

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A177040 Irregular triangle t(n,m) = binomial(m+1,n-m) read by rows floor((n+1)/2) <= m <= n.

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