{'fraction_non_alphanumeric': 0.04964539007092199, 'fraction_numerical': 0.0, 'mean_word_length': 4.578947368421052, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Implications of Spivey's Bell Number Formula

Recently, Spivey discovered a novel formula for B(n + m), where B(n + m) is the (n + m) th Bell number. His proof was combinatorial in nature. This paper provides a generating function proof of Spivey’s result. It also uses Spivey’s formula to determine a new formula for B(n). The paper concludes by extending all three identities to ordinary single variable Bell polynomials.