A363000 a(n) = numerator(R(n, n, 1)), where R are the rational polynomials R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).
1, 5, 19, 188, 1249, 125744, 283517, 303923456, 138604561, 599865008128, 118159023973, 7078040993755136, 155792758736921, 146303841678548271104, 294014633772018349, 64670474732430319157248, 752324747622089633569, 3224753626003393505960919040, 2507759850059601711479669
Offset: 0
Examples
a(n) are the numerators of the terms on the main diagonal of the triangle: [0] 1; [1] 1, 5/2; [2] 1, 7/2, 19/2; [3] 1, 11/2, 121/6, 188/3; [4] 1, 19/2, 95/2, 369/2, 1249/2; [5] 1, 35/2, 721/6, 1748/3, 35164/15, 125744/15; [6] 1, 67/2, 639/2, 3877/2, 18533/2, 76317/2, 283517/2;
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Programs
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Maple
# For better context we put A362998, A362999, A363000, and A363001 together here. R := (n, k, x) -> add(add(x^j*binomial(u, j)*(j+1)^n, j=0..u)/(u + 1), u=0..k): ### x = 1 -> this sequence for n from 0 to 7 do [n], seq(R(n, k, 1), k = 0..n) od; seq(R(n, n, 1), n = 0..9); A363000 := n -> numer(R(n, n, 1)): seq(A363000(n), n = 0..10); A363001 := n -> denom(R(n, n, 1)): seq(A363001(n), n = 0..20); A362999 := n -> denom(R(2*n+1, 2*n+1, 1)): seq(A362999(n), n = 0..11); A362998 := n -> add(R(2*n, k, 1), k = 0..2*n): seq(A362998(n), n = 0..9); ### x = -1 -> Bernoulli(n, 1) # for n from 0 to 9 do [n], seq(R(n, k,-1), k = 0..n) od; # seq(R(n, n, -1), n = 0..12); seq(bernoulli(n, 1), n = 0..12); ### x = 0 -> Harmonic numbers # for n from 0 to 9 do [n], seq(R(n, k, 0), k = 0..n) od; # seq(R(n, n, 0), n = 0..9); seq(harmonic(n+1), n = 0..9);
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