{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:11Z","timestamp":1753893851245,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We evaluate in closed form series of the type $\\sum u(n) R(n)$, with $(u(n))_n$ a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational\u00a0function. A typical example is:$$\\sum_{n \\geq 1} (-1)^{s_2(n)} \\frac{4n+1}{2n(2n+1)(2n+2)} = -\\frac{1}{4}$$where $s_2(n)$ is the sum of the binary digits of the integer $n$.\u00a0Furthermore closed formulas for series involving automatic sequences that are not strongly\u00a0$B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained;\u00a0for example, for integer $d \\geq 0$:$$\\sum_{n \\geq 0} \\frac{v(n)}{(n+1)^{2d+1}} = \\frac{\\pi^{2d+1} |E_{2d}|}{(2^{2d+2}-2)(2d)!}$$where $(v(n))_n$ is the $\\pm 1$ regular paperfolding sequence and $E_{2d}$ is an Euler\u00a0number.<\/jats:p>","DOI":"10.37236\/4630","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:30:31Z","timestamp":1578702631000},"source":"Crossref","is-referenced-by-count":0,"title":["Summation of Rational Series Twisted by Strongly $B$-multiplicative Coefficients"],"prefix":"10.37236","volume":"22","author":[{"given":"Jean-Paul","family":"Allouche","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Sondow","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2015,3,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p59\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:24:44Z","timestamp":1579256684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4630","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,3,6]]},"article-number":"P1.59"}}