A118921
Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n having first return to the x-axis at (2k,0) (n,k >= 1). (A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).
Original entry on oeis.org
2, 4, 2, 12, 4, 4, 40, 12, 8, 10, 140, 40, 24, 20, 28, 504, 140, 80, 60, 56, 84, 1848, 504, 280, 200, 168, 168, 264, 6864, 1848, 1008, 700, 560, 504, 528, 858, 25740, 6864, 3696, 2520, 1960, 1680, 1584, 1716, 2860, 97240, 25740, 13728, 9240, 7056, 5880, 5280
Offset: 1
T(3,2)=4 because we have uudd|ud, uudd|du, dduu|ud and dduu|du (first return to the x-axis shown by | ).
Triangle starts:
2;
4, 2;
12, 4, 4;
40, 12, 8, 10;
140, 40, 24, 20, 28;
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T:=(n,k)->2*binomial(2*k-2,k-1)*binomial(2*n-2*k,n-k)/k: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
A178046
Triangle t(n, m) = 2*binomial(n,m)^2 -A008292(n+1,m+1)^2 read by rows.
Original entry on oeis.org
1, 1, 1, 1, -8, 1, 1, -103, -103, 1, 1, -644, -4284, -644, 1, 1, -3199, -91004, -91004, -3199, 1, 1, -14328, -1418031, -5836256, -1418031, -14328, 1, 1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1, 1, -251876
Offset: 0
1;
1, 1;
1, -8, 1;
1, -103, -103, 1;
1, -644, -4284, -644, 1;
1, -3199, -91004, -91004, -3199, 1;
1, -14328, -1418031, -5836256, -1418031, -14328, 1;
1, -60911, -18428967, -243950711, -243950711, -18428967, -60911, 1;
1, -251876, -213392096, -7785232484, -24395306300, -7785232484, -213392096, -251876, 1;
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<< DiscreteMath`Combinatorica`
t[n_, m_] = 2*Binomial[n, m]^2 - Eulerian[n + 1, m]^2;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
A353596
Triangle read by rows, T(n, k) = [x^k] (-2)^n*GegenbauerC(n, -1/2, x).
Original entry on oeis.org
1, 0, 2, 2, 0, -2, 0, -4, 0, 4, -2, 0, 12, 0, -10, 0, 12, 0, -40, 0, 28, 4, 0, -60, 0, 140, 0, -84, 0, -40, 0, 280, 0, -504, 0, 264, -10, 0, 280, 0, -1260, 0, 1848, 0, -858, 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860, 28, 0, -1260, 0, 9240, 0, -24024, 0, 25740, 0, -9724
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 0, 2;
[2] 2, 0, -2;
[3] 0, -4, 0, 4;
[4] -2, 0, 12, 0, -10;
[5] 0, 12, 0, -40, 0, 28;
[6] 4, 0, -60, 0, 140, 0, -84;
[7] 0, -40, 0, 280, 0, -504, 0, 264;
[8] -10, 0, 280, 0, -1260, 0, 1848, 0, -858;
[9] 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860;
.
Unsigned antidiagonals |T(n+k, n-k)|:
[0] 1;
[1] 2, 2;
[2] 2, 4, 2;
[3] 4, 12, 12, 4;
[4] 10, 40, 60, 40, 10;
[5] 28, 140, 280, 280, 140, 28;
[6] 84, 504, 1260, 1680, 1260, 504, 84;
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g := n -> (-2)^n*GegenbauerC(n, -1/2, x):
seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);
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s={}; For[n=0,n<11,n++,For[k=0,kDetlef Meya, Oct 03 2023 *)
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