A176991
Triangle t(n,m) = binomial(n+m,m) - binomial(n-m,m), 1<=m<=n, read by rows.
Original entry on oeis.org
2, 2, 6, 2, 10, 20, 2, 14, 35, 70, 2, 18, 56, 126, 252, 2, 22, 83, 210, 462, 924, 2, 26, 116, 330, 792, 1716, 3432, 2, 30, 155, 494, 1287, 3003, 6435, 12870, 2, 34, 200, 710, 2002, 5005, 11440, 24310, 48620, 2, 38, 251, 986, 3002, 8008, 19448, 43758, 92378, 184756
Offset: 1
2;
2, 6;
2, 10, 20;
2, 14, 35, 70;
2, 18, 56, 126, 252;
2, 22, 83, 210, 462, 924;
2, 26, 116, 330, 792, 1716, 3432;
2, 30, 155, 494, 1287, 3003, 6435, 12870;
2, 34, 200, 710, 2002, 5005, 11440, 24310, 48620;
2, 38, 251, 986, 3002, 8008, 19448, 43758, 92378, 184756;
-
t[n_, m_] = Binomial[n + (m - 1), (m - 1)] - Binomial[n - (m - 1), (m - 1)];
Table[Table[t[n, m], {m, 2, n + 1}], {n, 1, 10}];
Flatten[%]
A327809
Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*Sum_{i=0..n} x^i*binomial(n, i)/(n+i+1).
Original entry on oeis.org
-1, 3, 2, -10, -15, -6, 35, 84, 70, 20, -126, -420, -540, -315, -70, 462, 1980, 3465, 3080, 1386, 252, -1716, -9009, -20020, -24024, -16380, -6006, -924, 6435, 40040, 108108, 163800, 150150, 83160, 25740, 3432, -24310, -175032, -556920, -1021020, -1178100, -875160, -408408, -109395, -12870
Offset: 0
Triangle begins:
-1;
3, 2;
-10, -15, -6;
35, 84, 70, 20;
-126, -420, -540, -315, -70;
462, 1980, 3465, 3080, 1386, 252;
-1716, -9009, -20020, -24024, -16380, -6006, -924;
...
Cf.
A046899 (Q(x) polynomials, up to sign).
Cf.
A001700 (1st column, up to sign),
A033876 (right diagonal, up to sign).
-
pol(n) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*sum(i=0, n, x^i*binomial(n, i)/(n+i+1));
row(n) = Vecrev(pol(n));
tabl(nn) = for (n=0, nn, print(row(n)));
Comments