The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 5
 1, 5, 19, 188, 1249, 125744, 283517, 303923456, 138604561, 599865008128, 118159023973, 7078040993755136, 155792758736921, 146303841678548271104, 294014633772018349, 64670474732430319157248, 752324747622089633569, 3224753626003393505960919040, 2507759850059601711479669 (list; graph; refs; listen; history; text; internal format)
 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 A102262 A123281 Adjacent sequences: A362997 A362998 A362999 * A363001 A363002 A363003 KEYWORD nonn,frac AUTHOR Peter Luschny, May 12 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 4 02:53 EST 2024. Contains 370522 sequences. (Running on oeis4.)