A253670 Square array read by ascending antidiagonals, T(n, k) = k!*[x^k](exp(x)*sum(j=0..n, C(2*n,j)*x^j)), n>=0, k>=0.
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 21, 7, 1, 1, 9, 43, 49, 9, 1, 1, 11, 73, 229, 89, 11, 1, 1, 13, 111, 529, 685, 141, 13, 1, 1, 15, 157, 1021, 3393, 1531, 205, 15, 1, 1, 17, 211, 1753, 8501, 12361, 2887, 281, 17, 1, 1, 19, 273, 2773, 18001, 63591, 32809, 4873, 369, 19, 1
Offset: 0
Examples
Square array starts: [n\k][0 1 2 3 4 5 6] [0] 1, 1, 1, 1, 1, 1, 1, ... [1] 1, 3, 5, 7, 9, 11, 13, ... [2] 1, 5, 21, 49, 89, 141, 205, ... [3] 1, 7, 43, 229, 685, 1531, 2887, ... [4] 1, 9, 73, 529, 3393, 12361, 32809, ... [5] 1, 11, 111, 1021, 8501, 63591, 272851, ... [6] 1, 13, 157, 1753, 18001, 169021, 1442173, ... The first few rows as a triangle: 1 1, 1 1, 3, 1 1, 5, 5, 1 1, 7, 21, 7, 1 1, 9, 43, 49, 9, 1 1, 11, 73, 229, 89, 11, 1
Crossrefs
Cf. A082545.
Programs
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Maple
T := (n,k) -> k!*coeff(series(exp(x)*add(binomial(2*n,j)*x^j,j=0..n), x, k+1), x, k): for n from 0 to 6 do lprint(seq(T(n,k), k=0..6)) od;
Formula
T(n,n) = A082545(n).