This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363000 #21 May 06 2025 11:27:00
%S A363000 1,5,19,188,1249,125744,283517,303923456,138604561,599865008128,
%T A363000 118159023973,7078040993755136,155792758736921,146303841678548271104,
%U A363000 294014633772018349,64670474732430319157248,752324747622089633569,3224753626003393505960919040,2507759850059601711479669
%N 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).
%C A363000 R(n, n, 0) are the (0-based) harmonic numbers, R(n, n, -1) are the Bernoulli numbers, and R(n, n, 1) is this sequence in its rational form.
%F A363000 Sum_{k=0..n} R(n, k, 0) = Sum_{j=0..n} (n-j+1)/(j+1) = (n+2)*Harmonic(n+1)-n-1.
%F A363000 Sum_{k=0..n} R(n, k,-1) = (n + 2 - 0^n) * Bernoulli(n, 1).
%F A363000 Sum_{k=0..2*n} R(2*n, k, 1) = A362998(n).
%F A363000 2*R(n, 1, 1) = A062709(n).
%e A363000 a(n) are the numerators of the terms on the main diagonal of the triangle:
%e A363000 [0] 1;
%e A363000 [1] 1, 5/2;
%e A363000 [2] 1, 7/2, 19/2;
%e A363000 [3] 1, 11/2, 121/6, 188/3;
%e A363000 [4] 1, 19/2, 95/2, 369/2, 1249/2;
%e A363000 [5] 1, 35/2, 721/6, 1748/3, 35164/15, 125744/15;
%e A363000 [6] 1, 67/2, 639/2, 3877/2, 18533/2, 76317/2, 283517/2;
%p A363000 # For better context we put A362998, A362999, A363000, and A363001 together here.
%p A363000 R := (n, k, x) -> add(add(x^j*binomial(u, j)*(j+1)^n, j=0..u)/(u + 1), u=0..k):
%p A363000 ### x = 1 -> this sequence
%p A363000 for n from 0 to 7 do [n], seq(R(n, k, 1), k = 0..n) od;
%p A363000 seq(R(n, n, 1), n = 0..9);
%p A363000 A363000 := n -> numer(R(n, n, 1)): seq(A363000(n), n = 0..10);
%p A363000 A363001 := n -> denom(R(n, n, 1)): seq(A363001(n), n = 0..20);
%p A363000 A362999 := n -> denom(R(2*n+1, 2*n+1, 1)): seq(A362999(n), n = 0..11);
%p A363000 A362998 := n -> add(R(2*n, k, 1), k = 0..2*n): seq(A362998(n), n = 0..9);
%p A363000 ### x = -1 -> Bernoulli(n, 1)
%p A363000 # for n from 0 to 9 do [n], seq(R(n, k,-1), k = 0..n) od;
%p A363000 # seq(R(n, n, -1), n = 0..12); seq(bernoulli(n, 1), n = 0..12);
%p A363000 ### x = 0 -> Harmonic numbers
%p A363000 # for n from 0 to 9 do [n], seq(R(n, k, 0), k = 0..n) od;
%p A363000 # seq(R(n, n, 0), n = 0..9); seq(harmonic(n+1), n = 0..9);
%Y A363000 Cf. A363001 (denominators), A362999 (odd-indexed denominators), A362998.
%Y A363000 Cf. A027611, A027612, A062709.
%K A363000 nonn,frac
%O A363000 0,2
%A A363000 _Peter Luschny_, May 12 2023