A363000 a(n) = numerator(R(n, n, 1)), where R are the rational poynomials 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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..18.

Formula

Sum_{k=0..n} R(n, k, 0) = Sum_{j=0..n} (n-j+1)/(j+1) = (n+2)*Harmonic(n+1)-n-1.

Sum_{k=0..n} R(n, k,-1) = (n + 2 - 0^n) * Bernoulli(n, 1).

Sum_{k=0..2*n} R(2*n, k, 1) = A362998(n).

2*R(n, 1, 1) = A062709(n).

Example

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;

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);

Crossrefs

Cf. A363001 (denominators), A362999 (odd-indexed denominators), A362998.

Cf. A027611, A027612, A062709.

Sequence in context: A082790 A145935 A024529 * A106991 A377556 A102262

Adjacent sequences: A362997 A362998 A362999 * A363001 A363002 A363003

Keyword

nonn,frac

Author

Peter Luschny, May 12 2023

Status

approved