A095789 -id:A095789 - cp's OEIS frontend

A216201 Square array T, read by antidiagonals : T(n,k) = 0 if n-k>=3 or if k-n>=4, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 0, 4, 6, 3, 0, 0, 4, 10, 9, 0, 0, 0, 0, 14, 19, 9, 0, 0, 0, 0, 14, 33, 28, 0, 0, 0, 0, 0, 0, 47, 61, 28, 0, 0, 0, 0, 0, 0, 47, 108, 89, 0, 0, 0, 0, 0, 0, 0, 0, 155, 197, 89, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 12 2013

			Square array begins:
1, 1, 1,  1,  0,   0,   0,   0,   0,   0, 0, 0, 0, ... row n = 0
1, 2, 3,  4,  4,   0,   0,   0,   0,   0, 0, 0, 0, ... row n = 1
1, 3, 6, 10, 14,  14,   0,   0,   0,   0, 0, 0, 0, ... row n = 2
0, 3, 9, 19, 33,  47,  47,   0,   0,   0, 0, 0, 0, ... row n = 3
0, 0, 9, 28, 61, 108, 155, 155,   0,   0, 0, 0, 0, ... row n = 4
0, 0, 0, 28, 89, 197, 352, 507, 507,   0, 0, 0, 0, ... row n = 5
0, 0, 0,  0, 89, 286, 638,1147,1652,1652, 0, 0, 0, ... row n = 6
...

E. Lucas, Théorie des nombres, Tome 1, Albert Blanchard, Paris, 1958, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A052975, A060557, A078038, A095789, A094790

T(n,n) = A052975(n).
T(n,n+1) = A060557(n).
T(n+1,n) = T(n+2,n) = A094790(n+1).
T(n,n+2) = T(n,n+3) = A094789(n+1).
Sum_{k, 0<=k<=n} T(n-k,k) = (-1)^n*A078038(n).

A095788 Table, read by antidiagonals, of iterated binomial transforms of A095148, which also forms the antidiagonal sums shift right.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 10, 15, 14, 1, 5, 17, 37, 51, 44, 1, 6, 26, 77, 150, 190, 155, 1, 7, 37, 141, 371, 656, 766, 605, 1, 8, 50, 235, 798, 1892, 3059, 3329, 2584, 1, 9, 65, 365, 1539, 4708, 10154, 15111, 15553, 11956, 1, 10, 82, 537, 2726, 10394, 28891

Offset: 0

Paul D. Hanna, Jun 06 2004

			From this table of iterated binomial transforms of A095148, the antidiagonal sums form the first row (A095148) shift left:
  1,  1,   2,    5,    14,     44,     155,      605,     2584, 11956, 59461, ...
  1,  2,   5,   15,    51,    190,     766,     3329,    15553, 77822, ...
  1,  3,  10,   37,   150,    656,    3059,    15111,    78840, ...
  1,  4,  17,   77,   371,   1892,   10154,    57077,   334993, ...
  1,  5,  26,  141,   798,   4708,   28891,   183953,  1212664, ...
  1,  6,  37,  235,  1539,  10394,   72350,   518505,  3821409, ...
  1,  7,  50,  365,  2726,  20840,  163091,  1306139, 10699288, ...
  1,  8,  65,  537,  4515,  38656,  337114,  2994701, 27094705, ...
  1,  9,  82,  757,  7086,  67292,  648539,  6344517, 63004248, ...
  1, 10, 101, 1031, 10643, 111158, 1175006, 12573713, ...
  ...

Cf. A095148 (first row), A095789 (diagonal).

cp's OEIS Frontend

A216201 Square array T, read by antidiagonals : T(n,k) = 0 if n-k>=3 or if k-n>=4, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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