# Thread:

1. ## Bernoulli numbers

Say we try to define the Bernoulli numbers using Faulhaber's formula...

$\sum_{k=0}^{m-1}k^n=\frac{1}{1+n}\sum_{k=0}^n\binom{n+1}{k}B_km^ {n+1-k}$

how do we show that the values $\{B_n\}$ are uniquely defined? That is, in the coefficients of $m^{n+1}, m^n, m^{n-1}\cdots$, $\frac{B_0}{1+n}, B_1, B_2 \frac{n}{2}, \cdots$ the values $B_0, B_1, B_2, \cdots$ will always be the same regardless of the power $n$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.6997509598731995, "perplexity_flag": "head"}