{
  "id": "enwiki-001371072699",
  "slug": "stirling-transform",
  "title": "Stirling transform",
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  "lead": "In combinatorial mathematics, the Stirling transform of a sequence { a n : n = 1, 2, 3, ... } of numbers is the sequence { b n : n = 1, 2, 3, ... } given by : b_n=\\sum_{k=1}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\} a_k , where \\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\} is the Stirling number of the second kind, which is the number of partitions of a set of size n into k parts. This is a linear sequence transformation. The inverse transform is : a_n=\\sum_{k=1}^n (-1)^{n-k} \\left[{n \\atop k}\\right] b_k , where (-1)^{n-k} \\left[{n\\atop k}\\right] is a signed Stirling number of the first kind, where the unsigned \\left[{n\\atop k}\\right] can be defined as the number of permutations on n elements with k cycles. Berstein and Sloane (cited below) state \"If a n is the number of objects in some class with points labeled 1, 2, ..., n (with all labels distinct, i.e. ordinary labeled structures), then b n is the number of objects with points labeled 1, 2, ..., n (with repetitions allowed).\" If : f(x) = \\sum_{n=1}^\\infty {a_n \\over n!} x^n is a formal power series, and : g(x) = \\sum_{n=1}^\\infty {b_n \\over n!} x^n with a n and b n as above, then : g(x) = f(e^x-1) . Likewise, the inverse transform leads to the generating function identity : f(x) = g(\\log(1+x)) .",
  "body": "In combinatorial mathematics, the Stirling transform of a sequence { a n : n = 1, 2, 3, ... } of numbers is the sequence { b n : n = 1, 2, 3, ... } given by b_n=\\sum_{k=1}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\} a_k , where \\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\} is the Stirling number of the second kind, which is the number of partitions of a set of size n into k parts. This is a linear sequence transformation. The inverse transform is a_n=\\sum_{k=1}^n (-1)^{n-k} \\left[{n \\atop k}\\right] b_k , where (-1)^{n-k} \\left[{n\\atop k}\\right] is a signed Stirling number of the first kind, where the unsigned \\left[{n\\atop k}\\right] can be defined as the number of permutations on n elements with k cycles. Berstein and Sloane (cited below) state \"If a n is the number of objects in some class with points labeled 1, 2, ..., n (with all labels distinct, i.e. ordinary labeled structures), then b n is the number of objects with points labeled 1, 2, ..., n (with repetitions allowed).\" If f(x) = \\sum_{n=1}^\\infty {a_n \\over n!} x^n is a formal power series, and g(x) = \\sum_{n=1}^\\infty {b_n \\over n!} x^n with a n and b n as above, then g(x) = f(e^x-1) . Likewise, the inverse transform leads to the generating function identity f(x) = g(\\log(1+x)) . ==See also== Binomial transform Generating function transformation List of factorial and binomial topics ==References== . Khristo N. Boyadzhiev, Notes on the Binomial Transform, Theory and Table, with Appendix on the Stirling Transform (2018), World Scientific.",
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    "title": "Stirling transform",
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    "publisher-place": "Dubai, United Arab Emirates",
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