{"paper":{"title":"Short note on the convolution of binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ant\\'onio Guedes de Oliveira, Rui Duarte","submitted_at":"2013-02-08T17:59:09Z","abstract_excerpt":"We know [Rui Duarte and Ant\\'onio Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n,i,j$ and for every real number $\\ell$, $$ \\sum_{i+j=n} \\binom{2i-\\ell}{i} \\binom{2j+\\ell}{j} = \\sum_{i+j=n}\\binom{2i}{i} \\binom{2j}{j}, $$ which is well-known to be $4^n$. We extend this result by proving that, indeed, $$ \\sum_{i+j=n} \\binom{ai+k-\\ell}{i} \\binom{aj+\\ell}{j} = \\sum_{i+j=n} \\binom{ai+k}{i} \\binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2100","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}