A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).
1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 5, 0, 1, 0, 0, 3, 15, 0, 1, 0, 0, 1, 8, 52, 0, 1, 0, 0, 0, 6, 25, 203, 0, 1, 0, 0, 0, 1, 25, 97, 877, 0, 1, 0, 0, 0, 0, 10, 91, 434, 4140, 0, 1, 0, 0, 0, 0, 1, 65, 322, 2095, 21147, 0, 1, 0, 0, 0, 0, 0, 15, 350, 1232, 10707, 115975, 0, 1, 0, 0, 0, 0, 0, 1, 140, 1702, 5672, 58194, 678570, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 0, 0, 0, 0, 0, ... 0, 2, 1, 0, 0, 0, 0, ... 0, 5, 3, 1, 0, 0, 0, ... 0, 15, 8, 6, 1, 0, 0, ... 0, 52, 25, 25, 10, 1, 0, ... 0, 203, 97, 91, 65, 15, 1, ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Programs
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PARI
T(n, k) = sum(j=0, n, stirling(n, k*j, 2));
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PARI
T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));
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PARI
Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!); T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);
Formula
For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.
For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).