A217779 -id:A217779 - cp's OEIS frontend

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 1, 5, 10, 10, 4, 0, 0, 6, 15, 20, 14, 0, 0, 0, 6, 21, 35, 34, 14, 0, 0, 0, 0, 27, 56, 69, 48, 0, 0, 0, 0, 0, 27, 83, 125, 117, 48, 0, 0, 0, 0, 0, 0, 110, 208, 242, 165, 0, 0, 0, 0, 0, 0, 0, 110, 318, 450, 407, 165

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
n=0: 1, 1,  1,  1,   1,   1,   0,   0,    0,    0,    0, 0, ...
n=1: 1, 2,  3,  4,   5,   6,   6,   0,    0,    0,    0, 0, ...
n=2: 1, 3,  6, 10,  15,  21,  27,  27,    0,    0,    0, 0, ...
n=3: 1, 4, 10, 20,  35,  56,  83, 110,  110,    0,    0, 0, ...
n=4: 0, 4, 14, 34,  69, 125, 208, 318,  428,  428,    0, 0, ...
n=5: 0, 0, 14, 48, 117, 242, 450, 768, 1196, 1624, 1624, 0, ...
...
Square array, read by rows, with 0 omitted:
...1,    1,     1,     1,     1,      1
...1,    2,     3,     4,     5,      6,      6
...1,    3,     6,    10,    15,     21,     27,     27
...1,    4,    10,    20,    35,     56,     83,    110,    110
...4,   14,    34,    69,   125,    208,    318,    428,    428
..14,   48,   117,   242,   450,    768,   1196,   1624,   1624
..48,  165,   407,   857,  1625,   2821,   4445,   6069,   6069
.165,  572,  1429,  3054,  5875,  10320,  16389,  22458,  22458
.572, 2001,  5055, 10930, 21250,  37639,  60097,  82555,  82555
2001, 7056, 17986, 39236, 76875, 136972, 219527, 302082, 302082
...
Triangle begins:
1
1, 1
1, 2,  1
1, 3,  3,  1
1, 4,  6,  4,  0
1, 5, 10, 10,  4,  0
0, 6, 15, 20, 14,  0, 0
0, 6, 21, 35, 34, 14, 0, 0
...

Cf. Similar sequences: A214846, A216054, A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235, A216236, A216238, A217257, A217315, A217593, A217765.

T(n,n+4) = T(n,n+5) = A094788(n+2).
T(n,n+3) = A217783(n).
T(n,n+2) = A217779(n).
T(n,n+1) = A081567(n).
T(n,n)   = A217782(n).
T(n+1,n) = A217778(n).
T(n+3,n) = T(n+2,n) = A094667(n+1).
Sum(T(n-k,k), k=0..n) = A217777(n).

A217782 Expansion of (1-x)(1-2x)(1-3x)/((1-5x+5x^2)(1-3x+x^2)).

Original entry on oeis.org

1, 2, 6, 20, 69, 242, 857, 3054, 10930, 39236, 141153, 508598, 1834641, 6623450, 23926334, 86468052, 312587197, 1130277914, 4087621545, 14784539846, 53478888618, 193456813508, 699850536281, 2531866279710, 9159810802849, 33139021206962, 119894215708662

Offset: 0

Philippe Deléham, Mar 24 2013

A diagonal of the square array A217770.

Index entries for linear recurrences with constant coefficients, signature (8, -21, 20, -5).

Cf. A217770, A217777, A217778, A217779.

a(n) = A217770(n,n).

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) -5*a(n-4) for n>=4, a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 20.

A217783 Expansion of (1-x)(1-2x)/((1-5x+5x^2)*(1-3x+x^2)).

Original entry on oeis.org

1, 5, 21, 83, 318, 1196, 4445, 16389, 60097, 219527, 799734, 2907800, 10558041, 38297573, 138819053, 502925211, 1821362830, 6594366404, 23870720757, 86396702117, 312668994969, 1131463798415, 4094241931526, 14814592074288, 53603587025713, 193949782284101

Offset: 0

Philippe Deléham, Mar 24 2013

A diagonal of the square array A217770.

Index entries for linear recurrences with constant coefficients, signature (8, -21, 20, -5).

Cf. A217770, A217777, A217778, A217779, A217782.

a(n) = A217770(n,n+3).

a(n) = 8*a(n-1) -21*a(n-2) + 20*a(n-3) -5*a(n-4) for n>=4, a(0) = 1, a(1) = 5, a(2) = 21, a(3) = 83.

cp's OEIS Frontend

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A217782 Expansion of (1-x)(1-2x)(1-3x)/((1-5x+5x^2)(1-3x+x^2)).

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A217783 Expansion of (1-x)(1-2x)/((1-5x+5x^2)*(1-3x+x^2)).

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cp's OEIS Frontend

A217770 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=4 or if k-n >= 6, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A217782 Expansion of (1-x)*(1-2x)*(1-3x)/((1-5x+5*x^2)*(1-3x+x^2)).

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A217783 Expansion of (1-x)*(1-2x)/((1-5x+5*x^2)*(1-3x+x^2)).

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A217782 Expansion of (1-x)(1-2x)(1-3x)/((1-5x+5x^2)(1-3x+x^2)).

A217783 Expansion of (1-x)(1-2x)/((1-5x+5x^2)*(1-3x+x^2)).