A103143
a(n) = a(n-1) + a(n-2) + 3*a(n-3), with a(0) = 1, a(1) = 0, a(2) = 1.
Original entry on oeis.org
0, 0, 1, 1, 2, 6, 11, 23, 52, 108, 229, 493, 1046, 2226, 4751, 10115, 21544, 45912, 97801, 208345, 443882, 945630, 2014547, 4291823, 9143260, 19478724, 41497453, 88405957, 188339582, 401237898, 854795351, 1821051995, 3879561040, 8264999088, 17607716113
Offset: 0
A108350
Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 32, 21, 6, 1, 1, 7, 31, 65, 65, 31, 7, 1, 1, 8, 43, 116, 161, 116, 43, 8, 1, 1, 9, 57, 189, 341, 341, 189, 57, 9, 1, 1, 10, 73, 288, 645, 842, 645, 288, 73, 10, 1, 1, 11, 91, 417, 1121, 1827, 1827, 1121, 417, 91
Offset: 0
Triangle rows begin
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 7, 4, 1;
1, 5, 13, 13, 5, 1;
1, 6, 21, 32, 21, 6, 1;
As a square array read by antidiagonals, rows begin
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 3, 7, 13, 21, 31, 43, ...
1, 4, 13, 32, 65, 116, 189, ...
1, 5, 21, 65, 161, 341, 645, ...
1, 6, 31, 116, 341, 842, 1827, ...
1, 7, 43, 189, 645, 1827, 4495, ...
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trgn(nn) = {for (n= 0, nn, for (k = 0, n, print1(sum(j=0, n-k, binomial(k,j)*binomial(n-j,k)*((j+1) % 2)), ", ");); print(););} \\ Michel Marcus, Sep 11 2013
A143685
Pascal-(1,9,1) array.
Original entry on oeis.org
1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1
Offset: 0
Square array begins as:
1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 11, 21, 31, 41, 51, 61, ... A017281;
1, 21, 141, 361, 681, 1101, 1621, ...
1, 31, 361, 1991, 5921, 13151, 24681, ...
1, 41, 681, 5921, 29761, 96201, 239241, ...
1, 51, 1101, 13151, 96201, 460251, 1565301, ...
1, 61, 1621, 24681, 239241, 1565301, 7272861, ...
Antidiagonal triangle begins as:
1;
1, 1;
1, 11, 1;
1, 21, 21, 1;
1, 31, 141, 31, 1;
1, 41, 361, 361, 41, 1;
1, 51, 681, 1991, 681, 51, 1;
1, 61, 1101, 5921, 5921, 1101, 61, 1;
1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;
Pascal (1,m,1) array:
A123562 (m = -3),
A098593 (m = -2),
A000012 (m = -1),
A007318 (m = 0),
A008288 (m = 1),
A081577 (m = 2),
A081578 (m = 3),
A081579 (m = 4),
A081580 (m = 5),
A081581 (m = 6),
A081582 (m = 7),
A143683 (m = 8), this sequence (m = 9).
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A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;
[A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021
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Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *)
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flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021
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